Mathematics / Logic Timeline

Mathematics began with the earliest records of attempts to quantify time. The Ishango Bone, a notched talley stick discovered at Ishango 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 one of the oldest known objects containing logical or mathematical carvings. 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.

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).

According to the theory about the origins of counting and writing developed by Denise Schmandt-Besserat, around 8000 BCE the Palaeolithic notched tallies representing the simplest form of counting — in one-to-one correspondence — were superseded by Neolithic clay tokens in various geometric forms suited for concrete counting invented in Mesopotamia. The significance of these tokens "as an operational device in Mesopotamian bureaucracy," was first grasped by archaeologist Pierre Amiet, teacher of Schand-Besserat in 1972 with respect to tokens found in Nuzi, an ancient Mesopotamian city southwest of Kirkuk in modern Al Ta'amim Governorate of Iraq, located near the Tigris river. (Schmandt-Besserat, Before Writing I [1992] ix.) 

"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 Shuruppaq, 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).

The Berlin Papyrus 6619, commonly known as the Berlin Papyrus, an ancient Egyptian papyrus document from the Middle Kingdom, was found at the ancient burial ground of Saqqara in the early 19th century CE.

"The papyrus is one of the primary sources of ancient Egyptian mathematical and medical knowledge, including the first known documentation concerning pregnancy test procedures, and is thus part of the medical papyri.

"The Berlin Papyrus contains a problem stated as "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."[4] The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equations in one unknown. In modern terms, the simultaneous equations x2 + y2 = 100 and x = (3/4)y reduce to the single equation in y: ((3/4)y)2 + y2 = 100, giving the solution y = 8 and x = 6" (Wikipedia article on Berlin Papyrus, accessed 12-29-2010).

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 in 1922 or 1923, and bequeathed it with the rest of his collection to Columbia University in 1936. 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.

"The most renowned of all mathematical cuneiform tablets since it was published in 1945, Plimpton 322 reveals that the Babylonians discovered a method of finding Pythagorean triples, that is, sets of three whole numbers such that the square of one of them is the sum of the squares of the other two. By Pythagoras' Theorem, a triangle whose three sides are proportional to a Pythagorean triple is a right-angled triangle. Right-angled triangles with sides proportional to the simplest Pythagorean triples turn up frequently in Babylonian problem texts; but if this tablet had not come to light, we would have had no reason to suspect that a general method capable of generating an unlimited number of distinct Pythagorean triples was known a millennium and a half before Euclid.  

"Plimpton 322 has excited much debate centering on two questions. First, what was the method by which the numbers in the table were calculated? And secondly, what were the purpose and the intellectual context of the tablet? At present there is no agreement among scholars about whether this was a document connected with scribal education, like the majority of Old Babylonian mathematical tablets, or part of a research project" (http://www.nyu.edu/isaw/exhibitions/before-pythagoras/items/plimpton-322/, accessed 11-23-2010).

Though the consensus may be that the tablet contains a listing of Pythagorean triples, Eleanor Robson pointed out that historical, cultural and linguistic evidence 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.

Yale YBC 7289, one of the few cuneiform tables to consist entirely of a geometrical diagram, shows that Babylonian scribes knew the Pythagorean Theorem and possessed a method of calculating accurate estimates of square roots. 

On the obverse, the scribe drew a square and its diagonals.

"According to Pythagoras' Theorem the length of the diagonal is the length of the side multiplied by the square root of 2. An accurate approximation of this quantity in sexagesimal notation is written along one diagonal. One side is labelled with its length, and the product of this number by the square root of 2 is also written along the diagonal" (http://www.nyu.edu/isaw/exhibitions/before-pythagoras/items/ybc-7289/, accessed 11-23-2010).

The tablet was acquired by 1944  by the Yale Babylonian Collection.

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.

Between 323 and 283 BCE mathematician Euclid of Alexandria, a teacher at the Alexandrian Library under the reign of Ptolemy I, wrote the Elements, “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 National Archaelogical Museum, 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 the island of Antikythera, Greece in 1900 or 1901, includes the only specimen preserved from antiquity of a scientifically graduated instrument. It may also be considered the earliest extant mechanical calculator. The device is displayed at the National Archaeological Museum of Athens, accompanied by a reconstruction made and donated to the museum by physicist and historian of science Derek de Solla Price.

"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 (accessed 01-2012) 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 (accessed 01-2012).

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 BCE. It is known that Archimedes invented a planetarium which calculated motions of the moon and the known planets. It is also believed that Archimedes wrote a manuscript, which did not survive, 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.

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♦ On December 12, 2010 a video showing the operation of a remarkable working reconstruction of the Antikythera Mechanism using plastic Lego parts could be viewed on the blog of Make Magazine.

Hellenistic astronomer, geographer, and mathematician, Hipparchos of Rhodes, produced 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, who was probably born Nicaea (now Iznik, Turkey) and probably died on the island of Rhodes. 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 introduced 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 was taken in China during the Han Dynasty. At that time there were 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)."

About 270 CE Neoplatonic philosopher Porphyry of Tyre (Πορφύριος, Porphyrios) published his Introduction (εἰσαγωγή, Isagoge) to logic, perhaps while he was in Rome. In this work, which was translated into Latin by Anicus Manlius Severinus Boëthius, and remained a disseminated and copied text throughout the Middle Ages, Porphyrios reframed the predictables (praedicamenta) defined by Aristotle in his Organon into a list of five classes: genus (genos), species (eidos), difference (diaphora), property (idion), and accident (sumbebekos). From these Porphyrios created the scala praedicamentalis, or Porphyrian Tree (Tree of Porphyry, Arbor Porphyriana).

Porphyry's introduction was the most successful work of its kind ever published. Translated into many languages, for 1500 years every student read it as the first text on its subject. As a result, its influence was immense in philosophy and logic, and in the organization of knowledge, and its visualization in arborial form.

"Expanding on Aristotle's Categories and visually alluding to a tree's trunk, Porphyry's structure reveals the idea of a layered assembly in logic. It is made of three columns of words, where the central column contains a series of dichomatous divisions between genus and species, whcih derive from the supreme genus, Substance. Even through Porphyry himself never drew such an illustration—his original tree was purely textual in nature—the symbolic tree of Porphyry was frequently represented in medieval and Renaissance works on logic and set the stage for theological and philosophical developments by scholars throughout the ages. It was also, as far as we know, the earliest metaphorical tree of knowledge" (Lima, Visual Complexity: Mapping Patterns of Information [2011] 28).

The standard of edition of Porphryios's text is Porphyri. Introduction, Translated with a Commentary by Jonathan Barnes (Oxford, 2003). Of this work of xxxi, 415pp., only the first 19 pages consist of the translation of Porphyri's brief text. From it we learn that

"as a young man he [Porphyry] removed to Athens, where he studied rhetoric, mathematics and philosophy with Longinus, the 'living library and walking museum'. . .In 263 he migrated to Rome and joined the magic circle of Plotinus.. .. He became a fevern and favoured acolyte of Plotinus . But he remained with him for no more than five years; in 268 he fell sick with a melancholy and Plotinus urged him south to Sicily for his health's sake.

"In 270 Plotnius died. Later, Porphyry returned to Rome, where he lectured on his master's philosophy—and where, in 301, he made public his edition of Plotinus' Enneads. When, and for how long, he was back in Rome we cannot tell; nor is it known when he visited North Africa (where he stayed long enough to befriend a partridge). Late in life he married (and not for love). In a letter to his wife Marcella, he explains that he must leave her to look after the 'interests of the pagans...: some have inferred that Porphyry, an enemy of Christianity, was summoned to the imperial capital to advise the persecuting Emperor Diocletian. 

"The date and place of his death are unknown" (Barnes p. x).

On charges of treason, Theodoric the Great, Ostrogothic ruler of Italy, executed 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 taken as a date for 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, used a true zero in tables alongside Roman numerals, but he used 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."

Indian mathematician and astronomer of Bhinmal (भीनमाल), a town in the Jalore District of Rajasthan, India,  Brahmagupta wrote 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 was widely used in India, and the concept of zero (represented by a dot) was known.

Balthild, widow of Clovis II, and her son Clotaire III, founded 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 [25 km southwest of Aleppo]), 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.

In 1725 the Venerable Bede, a monk at the Northumbrian monastery of Saint Peter at Monkwearmouth, wrote 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 De computo, vel loquela per gestum digitorum appears to have made its first appearance in print In Hoc in volumine haec continentur M. Val. Probus de notis Roma. ex codice manuscript castigatior . . . , ed. Giovanni Tacuino published in Venice by the editor,Tacuino, who was also a printer, in 1525. The editio princeps of De temporum ratione was published by Sichardus in 1529, four years after Tacuino issued his edition. Portions of De temporum ratione appeared in print as early as 1505, but these do not appear to have included the section on finger-reckoning. Smith, in his Rara arithmetica, stated that the 1522 edition of Johannes Aventinus’s Abacus atque vetustissima, veterum latinorum per digitos manusque numerandi contains a description of Bede’s finger-reckoning; however, this may be an error, since there was no record of this edition in OCLC or the Karlsruhe Virtual Catalogue when we searched the database in March 2013. Smith himself described only the 1532 edition of Aventinus’s work (see Rara arithmetica, pp. 136-138).

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

The second Abbassid Caliph, Abu Ja'far Al-Mansur, founded the city of Baghdad. There he founded a palace library, which evolved into The House of Wisdom. The library was originally concerned with translating and preserving Persian works, first from Pahlavi (Middle Persian), then from Syriac and eventually Greek and Sanskrit.

"The House of Wisdom acted as a society founded by Abbasid caliphs Harun al-Rashid and his son al-Ma'mun who reigned from 813-833 CE. Based in Baghdad from the 9th to 13th centuries, many of the most learned Muslim scholars were part of this excellent research and educational institute. In the reign of al-Ma'mun, observatories were set up, and The House was an unrivalled centre for the study of humanities and for sciences, including mathematics, astronomy, medicine, chemistry, zoology and geography. Drawing on Persian, Indian and Greek texts—including those of Pythagoras, Plato, Aristotle, Hippocrates, Euclid, Plotinus, Galen, Sushruta, Charaka, Aryabhata and Brahmagupta—the scholars accumulated a great collection of knowledge in the world, and built on it through their own discoveries. Baghdad was known as the world's richest city and centre for intellectual development of the time, and had a population of over a million, the largest in its time.The great scholars of the House of Wisdom included Al-Khawarizmi, the "father" of algebra, which takes its name from his book Kitab al-Jabr" (Wikipedia article on House of Wisdom, accessed 12-01-2008).

The House of Wisdom flourished until it was destroyed by the Mongols in the sacking of Baghdad in 1258.

Regarding the transmission of Hindu numbers to the Arabs, al-Qifti's The History of the Learned Men 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, developed 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, a scholar in the House of Wisdom in Baghdad, published 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. This 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 furnished 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 (Kayseri, Turkey) for 14 nomismata (gold coins).

"The hand of Stephanus is pure minuscule; Arethas added the scholia and some additional matter 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.

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 was a scholar, teacher, tutor and counsellor to Otto III before being elevated to the papacy as Sylvester II (or Silvester II) from 999 till his death in 1002. He was 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).

Between 1150 and 1175 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 Córdoba 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, who taught in Aleppo and Mosul, originated 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 appeared in China, called suanpan in Chinese. On each rod this abacus had 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) [a native of Dongwu (present Zhucheng, Shandong)] 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, wrote 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."

In contrast to Euclid's Elements, which were written at the Royal Library of Alexandria, and widely disseminated, the writings of the Greek mathematician, physicist, engineer, inventor, and astronomer Archimedes were not widely known in antiquity. Survival of their texts was due to interest in Archimedes' writings at the Byzantine capital of Constantinople from the sixth through the tenth centuries.

"It is true that before that time individual works of Archimedes were obviously studied at Alexandria, since Archimedes was often quoted by three eminent mathematicians of Alexandria: Hero, Pappus, and Theon. But it is with the activity of Eutocius of Ascalon, who was born toward the end of the fifth century and studied at Alexandria, that the textual history of a collected edition of Archimedes properly begins. Eutocius composed commentaries on three of Archimedes' works: On the Sphere and the Cylinder, On the Measurement of the Circle, and On the Equilibrium of Planes. These were no doubt the most popular of Archimedes' works at that time. . . . The works of Archimedes and the commentaries of Eutocius were studied and taught by Isidore of Miletus and Anthemius of Tralles, Justinian's architects of Hagia Sophia in Constantinople. It was apparently Isidore who was responsible for the first collected edition of at least the three works commented on by Eutocius as well as the commentaries. Later Byzantine authors seem gradually to have added other works to this first collected edition until the ninth century when the educational reformer Leon of Thessalonica produced the compilation represented by Greek manuscript A (adopting the designation used by the editor, J. L. Heiberg).  Manuscript A contained all of the Greek works now known excepting On Floating Bodies, On the Method, Stomachion, and The Cattle Problem. This was one of the two manuscripts available to William of Moerbeke when he made his Latin translations in 1269.  It was the source, directly or indirectly, of all of the Renaissance copies of Archimedes. A second Byzantine manuscript, designated as B, included only the mechanical works: On the Equilibrium of Planes, On the Quadrature of the Parabola and On Floating Bodies (and possibly On Spirals).  It too was available to Moerbeke. But it disappears after an early fourteenth-century reference. Finally we can mention a third Byzantine manuscript, C, a palimpsest whose Archimedean parts are in a hand of the tenth century. It was not available to the Latin West in the Middle Ages, or indeed in modern times until its identification by Heiberg in 1906 at Constantinople (where it had been brought from Jerusalem)" (Marshall Clagett, "Archimedes," Dictionary of Scientific Biography I [1970] 223).

Transmission of Archimedes' writings to the west was largely dependent upon the translation into Latin of most of the Archimedean texts in manuscripts A and B by the Flemish Dominican William of Moerbeke (Willem van Moerbeke) in 1269.  These manuscripts had passed into the Pope's library from the collection of the Norman kings of the Two Sicilies.  Moerbeke's translations of the two manuscripts were not without errors, but they presented the texts in an understandable way. The holograph of Moerbeke's translation survives in the Vatican Library (MS Vat. Ottob. lat. 1850). It was not widely copied. Manuscripts A and B no longer survive.

"In the fifteenth century, knowledge of Archimedes in Europe began to expand. A new latin translation was made by James of Cremona in about 1450 by order of Pope Nicholas V. Since this translation was made exclusively from manuscript A, the translation failed to include On Floating Bodies, but it did include the two treatises in A omitted by Moerbeke, namely The Sand Reckoner and Eutocius' Commentary on the Measurement of the Circle. It appears that this new translation was made with an eye on Moerbeke's translation. . . . There are at least nine extant manuscripts of this translation, one of which was corrrected by Regiomontanus and brought to Germany about 1468. . . . Greek manuscript A itself was copied a number of times. Cardinal Bessarion had one copy prepared between 1449 and 1468 (MS E). Another (MS D) was made from A when it was in the possession fo the well-kinown humanist George [Giorgio] Valla. The fate of A and its various copies has been traced skillfully by J. L. Heiberg in his edition of Archimedes' Opera. The last known use of manuscript A occurred in 1544, after which time it seems to have disappeared.  The first printed Archimedean materials were in fact merely latin excerpts that appeared in George Valla's De expetendis et fugiendis rebus opus (Venice, 1501) and were based on his reading of manuscript A. But the earliest actual printed texts of Archimedes were the Moerbeke translations of On the Measurement of the Circle and On the Quadrature of the Parabola (Teragonismus, id est circuli quadratura etc.) published from the Madrid manuscript by L.[uca] Gaurico (Venice, 1503). In 1543 also at Venice N.[iccolo] Tartaglia republished the same two translations directly from Gaurico's work, and in addition, from the same Madrid manuscript, the Moerbeke translations of On the Equilbrium of Planes and Book I of On Floating Bodes (leaving the erroneous impression that he had made these translations from a Greek manuscript, which he had not since he merely repeated the texts of the Madrid manuscript, with virtually all their errors.) . . . The key event, however, in the further spread of Archimedes was the aforementioned editio princeps of the Greek text with the accompanying Latin translation of James of Cremona at Basel in 1544. . . ." Clagett, op. cit., 228-229).

For the editio princeps the editor Thomas Gechauff, called Venatorius (d. 1551), was able to use the above-mentioned manuscript of James of Cremona's (Jacopo da Cremona's) Latin translation corrected by Regiomontanus, which included the commentaries of Eutocius of Ascalon. For the Greek text Gechauff used a manuscript which had been acquired in Rome by humanist Willibald Pirckheimer, and is preserved today today in Nuremberg City Library.

Existence of a reliable Greek and Latin edition made the texts available to a wider range of scholars, exerting a strong influence on mathematics and physics in the sixteenth century. "One of the imortant effects of that influence can be seen in Kepler's Astronomia nova, in which Archimedes's so-called 'exhaustion procedure' was applied to the measurement of time elapsed between any two points in Mars's orbit" (Hook & Norman, Haskell F. Norman Library of Science and Medicine [1991] no. 61).

♦ After disappearing into a European private collection in the early twentieth century, the third key record of Archimedes' texts discussed above, the tenth century Byzantine manuscript C, known as the Archimedes Palimpsest, re-appeared at a Christie's auction in New York on October 28, 1998, where it was purchased by an anonymous private collector in the United States. Since then it has been made widely available to scholars, and has been the subject of much research.

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.

Around 1305 Majorcan writer and philosopher Ramon Llull (Lull) published 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. Containing images representing Aztec measurement of days, months, and cosmic cycles, the stone was completed during the 52 year period between 1427 and 1479 CE. 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).

In 1434 Michael of Rhodes, a Venetian galley commander, wrote 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.

On May 25, 1482 printer Erhard Ratdolt of Venice issued 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 published 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.

Between November 10 and 20, 1494 Fra Luca Bartolomeo de Pacioli published 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 points out the very unusual aspect of the edition that two re-issues of the first edition exist with some sheets reprinted. One of these is thought to date after 1509 and another after 13 August 1502. Nevertheless, these re-issues bear the original publication date.  

English Scholastic, church leader, diplomat, administrator and royal adviser, then Bishop of London, Cuthbert Tunstall (Tonstall, Tonstal) published De arte supputandi libri quattuor in London at the press of Richard Pynson. Based on the Summa de arithmetica of Luca Bartolomeo de Pacioli, this was the first printed work published in England that was devoted exclusively to mathematics. Its woodcut title was engraved by Hans Holbein the Younger.

"In the dedicatory epistle Tonstall states that in his dealing with certain goldsmiths he suspected that their accounts were incorrect, and he therefore renewed his study of arithmetic so as to check their figures. On his appointment to the See of London he bade farewell to the sciences by printing this book in order that others might have the benefit of work which he had prepared for his own use. The treatise is in Latin, and, although it was written for the purpose of supplying a practical handbook, is very prolix and was not suited to the needs of the mercantile class. It is confessedly based upon Italian models, and it is apparent that Tonstall must have known, from his reidence in Padua and his various visits to Italy, the works of the leading Italian writers. The book includes many business applications of the day, such as partnership, profit and loss, and exchange. It also includes the rule of false, the rule of three, and numerous applications of these and other rules. It is, however, the work of a scholar and a classicist rather than a business man.

"The word 'supputandi,' in the title, was not uncommon at that time. Indeed there was some tendency to use the 'supputation' for arithmetic and to speak of calculations as 'supputations.'

"Tonstall dedicates the work to his friend Sir Thomas More, whose talented daughter Erasmus addressed as 'Margareta Ropera Britanniae tuae decus,' —ornament of thine England. More speaks of Tonstall in the opening lines of his Utopia: 'I was colleague and companion to that incomparable man Cuthbert Tonstal, whom the king with such universal applause latelly made Master of the Rolls; but of whom I will say nothing; not because I fear that the testimony of a friend will be suspected, but rather because his learning and virtues are too great forme to do them justice, and so well known, that they need not by commendation unless I would, according to the prover, 'Show the sun with a lanthorn.' . . . ." (Smith, Rara Arithmetica I [1908] 132-34).

In September 1533 Printer Johannes Herwagen (Hervagius) of Basel published 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 in the history of graphic design 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 published 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 unusually extended delay between the publication of the Copernican theory and its acceptance by the scientific community, for many years historians 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.

French painter, sculptor, etcher, engraver, and geometrician, Jean Cousin the Elder, published 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). 

In 1570 English merchant, and later Lord Mayor of London Henry Billingsley issued in London 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 from the well-known Latin translation by Adelard of Bath and Campanus of Novara.  In the nineteenth century 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, preserved at Princeton University Library.  Billingsley's copy is 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. 

"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 English translation of Euclid 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) published in Paris 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 issued a papal bull, Inter gravissimas, the founding document of the Gregorian calendar. It was printed on March 1, 1562.

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 from his own house in Padua,printed by Peitro Marinelli, 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.

Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun John Napier published in Edinburgh his Mirifici logarithmorum canonis descriptio, announcing his invention of logarithms, with the goal of increasing calculating speed and reducing drudgery.

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

"He [Napier] wrote that the multiplication and division of great numbers is troublesome, involving tedious expenditure of time, and subject to "slippery errors." His tables reduced these difficulties to simple addition and subtraction, and won immediate recognition. A set of Napier’s bones are usually made of boxwood or ivory and often contained in a box or case that would fit in a pocket. A set usually contains 10 rods, plus extras representing squares and cubes.  

"Use. Addition is accomplished by reading the appropriate bones along the diagonal. To obtain a product of 224 x 44, the rods 2, 2, and 4 are put alongside each other, and the result is read off by combining the numbers in the fourth row -- 0/8, 0/8, 1/6 -- for the correct answer 896. This is repeated and the two products added together to give 9856. The bones are sometimes associated with an abacus to provide a store in the multiplication process" (Gordon Bell's website, accessed 10-12-2011).

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

In 1628 Adriaan Vlacq, a bookseller, publisher, and human computer, computed and issued the first complete set of modern logarithms in Gouda through Petrus Rammaseyn printers. Four years earlier, in 1624, English mathematician Henry Briggs had published Arithmetica logarithma sive logarithmorum chiliades triginta, pro numeris naturali serie crescentibus ab unitate 20,000 et a 90,000 ad 100,000 changing the original logarithms invented by John Napier into common (base 10) logarithms. In 1626 Dutch surveyer and teacher of mathematics Ezechiel de Decker contracted with Vlacq for the publication of several translations of books by John Napier, Edmund Gunter and Henry Briggs. A first book was published in 1626, with several translations done by Vlacq. A second book was made of the logarithms of the first 10000 numbers from Briggs' Arithmetica logarithmica published in 1624. The logarithms were shortened to 10 places. In 1627, De Decker's Het Tweede deel van de Nieuwe telkonst  was published, containing the logarithms of all numbers from 1 to 100000, to 10 places, much of which had been computed by Vlacq. Only very few copies of this book are known and its publication was apparently stopped or delayed.This Tweede deel of 1627 was the first complete table of decimal logarithms. 

In 1628 Vlacq republished the 10 decimal place logarithm tables as Arithmetica logarithma sive logarithmorum chiliades tentum, pro numeris naturali serie crescentibus ab unitate ad 100000. He appears to have had a connection with the Gouda firm of Petrus Rammaseyn and it is this firm that published the work, this time under Vlacq's name. A French translation, Arithmetique logarithmetique, ou, La construction et usage d'une table contenant les logarithms de tous les nombres depuis l'unité jusque 100000 by Vlacq was also published by Petrus Rammaseyn at almost the same time.

In a letter to theologian, philosopher, and mathematician Marin Mersenne, philosopher, mathematician and physicist René Descartes proposed 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).

English priest and mathematician William Oughtred invented the circular form of slide rule. He published Circles of Proportion and the Horizontal Instrument in London in 1632 describing slide rules and sundials.

French philosopher, mathematician, and scientist René Descartes issued 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.

French lawyer and amateur mathematician Pierre de Fermat owned a copy of the 1621 Paris edition of the Arithmetica by the ancient Greek mathematician Diophantus, edited by Claude Gaspard Bachet de Méziriac, and was in the habit of noting his own number theory propositions in the margins of the book. In 1637 Fermat made a marginal note next to one of the problems put forth by Diophantus, stating, in essence, that equations of the form xn + yn = zn have no whole-number solutions when n is greater than 2. In his note Fermat stated that he had found a truly marvelous proof (demonstratio mirabilis), which would not fit into the narrow margin of the book.

Fermat died in 1665 without revealing his proof known as Fermat's Last Theorem. In 1670 Fermat’s son published a second edition of Bachet’s edition of Diophantus from the press of Bernard Bosc in Toulouse that incorporated all of Fermat’s marginal notes and propositions, from which Fermat's Last Theorem became widely known. Today scholars doubt that he actually achieved it.

Most of Fermat’s propositions were proved during the 18th century, but the Last Theorem remained a stumbling block for succeeding generations of mathematicians, and by the early 19th century it had gained a reputation as perhaps the world’s most baffling mathematical mystery. “Simple, elegant, and [seemingly] impossible to prove, Fermat’s Last Theorem captured the imaginations of amateur and professional mathematicians for over three centuries. For some it became a wonderful passion. For others it was an obsession that led to deceit, intrigue, or insanity” (Aczel).

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 777.

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

"Use. The dials show the French monetary unit, the livre, which was divided into 12 deniers, each subdivided into 20 sols. The essential part of the machine was its decimal carry; each toothed wheel moved forward one unit (one-tenth of a revolution on each wheel except those of deniers and sols) when the previous wheel had completed one revolution. Subtraction was based on complementary numbers that could be revealed by moving the strip at the top of the calculator" (Gordon Bell's website, accessed 10-12-2011).

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.

German Jesuit scientist Gaspard Schott's posthumous Organum Mathematicum was published in Nuremberg, in which Schott described his “mathematical organ,” and his calculating machine based on Napier’s rods.

In Dissertations academiques. . . avec un discours sur. . . un cylindre arithmetique published in Paris Pierre Petit described an arithmetic cylinder, which he said was more affordable and easier to use than Pascal’s Pascaline.

English diplomat, mathematician and inventor Samuel Morland published in London The Description and Use of Two Arithmetic Instruments, the first monograph on a calculating machine published in English. The book described modifications to the Pascaline.

Gottfried Wilhelm Leibniz made 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 published 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 Gottfried Wilhelm Leibniz, Bibliothek Niedersächsische 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 in 1966 to commemorate the 250th anniversary of Leibniz's death as Herrn von Leibniz' Rechnung mit Null und Eins. That book included facsimiles of Leibniz's "Explication de l'arithmétique binaire" (1705), his two letters to Johann Christian Schulenberg on binary arithmetic (March 29 and May 17, 1698), published in the Opera Omnia of 1768, and historical articles and German translations.

Gottfried Wilhelm Leibniz published "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 issued from Leipzig. 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.

Carter & Muir, Printing and the Mind of Man (1967) no. 160.

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1326.  

♦ In April 2012 I learned that there are three issues of this publication involving two different settings of type, and two different versions of the copperplate geometrical diagram. An early issue, incorporating numerous mathematical errors in the typesetting on p. 467, was included in the Norman library.  It is illustrated in volume two of Christie's auction catalogue (1998) lot 613.  A different, and presumably later printing with the errors corrected on p. 467, is illustrated by Horblit, One Books Famous in Science (1964) no. 66a.  A third issue, either before or after that in the Norman library, but prior to that described by Horblit, was reported by Dieter Schierenberg BV in 2011. That issue incorporates the earlier state of p. 467 but with the addition of "M. Oct." at the top of the plate under the plate number.

Isaac Newton published Philosophia naturalis principia mathematica in London through the efforts and expense of astronomer Edmond Halley.

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.

Bookseller and city counceller in Emmerich, Cornelis a Beughem issued 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 contained a bibliography of atlases.

Bibliotheca mathematica was one of a series of bibliographies Beughem issued through the Amsterdam firm Janssonius-Waesberghe, listing books published throughout Europe in the relevant subject area during the second half the seventeenth century in any language, whether first or revised editions. Beughem's bibliographies were distinguished from earlier bibliographies by their arrangement by author, and by their limited chronological coverage to the present and the immediate past. Bibliographia mathematica followed bibliographies by Beughem of law and politics (1680) and medicine and physics (1681). 

Breslauer & Folter, Bibliography: Its History and Development (1984) no. 84.

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

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.

In 1710 German mathematician and philosopher Gottfried Wilhlem Leibniz published "Brevis descriptio Machinae Arithmeticae, cum Figura" in Miscellanea Berolinensis (1710) 317-19, fig. 73. This was the first description of Leibniz's stepped-drum calculator, or stepped reckoner. Because Leibniz had only two working examples of the machine made, and one was lost, his invention of the stepped reckoner was primarily known through this and other publications.

Isaac Newton published 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 issued 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.

French Hugenot mathematician and demographer exiled in England, Abraham de Moivre published Annuities upon Lives: Or, the Valuation of annuities upon any Number of lives; as also, of Reversions.

Using the mortality statistics gathered by Edmond Halley in the 1690s, Moivre formulated the theory of annuities, deriving his formulas from a postulated uniform rate of mortality and constant rates of interest on money.  "Here one finds the treatment of joint annuities on several lives, the inheritance of annuities, problems about the fair division of the costs of a tontine, and other contracts in which both age and interest on capital are relevant.  This mathematics became a standard part of all subsequent commercial applications in England" (Dictionary of Scientific Biography).

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1530.

In Gulliver's Travels Jonathan Swift describes a fictional device called The Engine, which generates permutations of word sets. It is possibly the earliest literary reference to a fictional device resembling aspects of a modern computer.  Though Swift does not reference the medieval Ars generalis ultima (Ars magna) of the Spanish philosopher Ramon Llull (Lull), called by Leibniz, the ars combinatoria in Leibnitz's De arte combinatoria, the passage is considered a parody of Llull's 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."

In 1736 Swiss German mathematician and physicist Leonhard Euler, working at the Imperial Russian Academy of Sciences in St. Petersburg, published "Solutio problematis ad geometriam situs pertinentis," Commentarii Academiae Scientiarum Imperialis Petropolitanae 8 (1736) 128-40. This negative solution to the Seven Bridges of Königsberg problem represented the beginning of graph theory, topology and network science.

An extended English translation of Euler's paper appeared in Biggs, Lloyd & Wilson, Graph Theory 1736-1936 (1977) 1-20.

Lima, Visual Complexity: Mapping Patterns of Information (2011) 74-75.

In 1742 German mathematical historian and theologian Johann Christoph Heilbronner issued Historia matheseos universae a mundo condito ad seculum from Leipzig.

Heilbronner’s work, of which an abbreviated German edition ("Erster Theil") was published in 1739, was the first to use the term “mathematical history.” It is one of the earliest histories of any science, predating Jean-Etienne Montucla's Histoire des mathématiques, which began publication in 1758. Perhaps mathematics was the first science to be studied historically since it was the first scientific subject required in all curricula—a requirement since the medieval quadrivium, and perhaps earlier. 

Heilbronner’s complete Latin edition, containing about 1000 pages (roughly 800 more pages than the abbreviated German edition) contains chapters on mathematics and its uses, 602 biographies of famous mathematicians, bio-bibliographies of mathematical textbook writers, a chapter on Chinese mathematics, and a special study of arithmetic, including sections on arithmetical writers and even arithmetical poetry and divination. Of particular interest is a section listing mathematical manuscripts in important Italian, French, German and British libraries; some of the materials cited here may no longer be extant

French mathematician and statistician Antoine Deparcieux issued in Paris 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."

J. Norman (ed) Morton's Medical Bibliography, 5th ed. (1991) no. 1691.1

In 1750 the Neopolitan polymath and inventor Raimondo di Sangro, Prince of Sansevero, issued Lettera apologetica dell'esercitato accademico della Crusca contenente la Difesa del Libro Intitolato Lettere d'una Peruana per rispetto alla supposizione de'Quipu from the press of Gennaro Morelli of Naples. This work, printed in color using a polychromatic printing process invented by the Prince, was the first extensive treatise on the Peruvian knot-based counting language, the Quipu.  

Quipu used a decimal positional system: a knot in a row farthest from the main strand represented one, next farthest ten, etc.; the absence of knots on a cord implied zero. The colors of the cords, the way the cords are connected together, the relative placement of the cords, the spaces between the cords, the types of knots on the individual cords, and the relative placement of the knots are all important parts of the recording system. ‘Quipucamayocs,’ the accountants of the Inca Empire, created and deciphered the Quipu knots, and were also capable of performing simple mathematical calculations such as adding, subtracting, multiplying, and dividing. Quipu accounts were kept by court historians in Peru that covered hundreds of years of history, but after the Conquest, the Spaniards began to resent having this second set of record-keepers contradict them. The Quipu was classified as idolatrous at the Third Council of Lima (1581-3), many examples were destroyed.  Thus, by the time Raimondo di Sangro published his book the Quipu was no longer practiced, and attempting to understand the language was a research project in cryptanalysis.

"To date, no link has yet been found between a quipu and Quechua, the native language of the Peruvian Andes. This suggests that quipus are not a glottographic writing system and have no phonetic referent. Frank Salomon at the University of Wisconsin has argued that quipus are actually a semasiographic language, a system of representative symbols—such as music notation or numerals—that relay information but are not directly related to the speech sounds of a particular language. The Khipu Database Project (KDP), begun by Gary Urton, may have already decoded the first word from a quipu—the name of a village, Puruchuco, which Urton believes was represented by a three-number sequence, similar to a ZIP code. If this conjecture is correct, quipus are the only known example of a complex language recorded in a 3-D system. (Wikipedia article on Quipu, accessed 04-07-2013).

In 1755 English mathematician Thomas Simpson published "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.  Simpson's paper was "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 "An Essay Towards Solving a Problem in the Doctrine of Chances" by English clergyman and mathematician Thomas Bayes was 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 sanctioned 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 who produced them, working by hand and separated geographically in an early example of organized but distant collaboration.

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.

French mathematician and engineer Gaspard Clair François Marie Riche de Prony, Engineer-in-Chief of the École Nationale des Ponts et Chaussées, undertook, beginning in 1793, the production of logarithmic and trigonometric tables for the French Cadastre. He was asked to produce the tables by the French National Assembly, which, after the French Revolution, wanted to bring uniformity to the multiple measurements and standards used throughout the nation. The tables and their production were vast, with values calculated to between fourteen and twenty-nine decimal places.

Inspired by Adam Smith's Wealth of Nations, de Prony produced the tables through the systematic division of labor, bragging that he could manufacture logarithms as easily as one could manufacture pins. At the top of the organizational hierarchy were scientists and mathematicians who devised the formulas. Next were workers who created the instructions for doing the calculations. At the bottom were about ninety human computers who were not trained in mathematics, but who followed instructions very carefully. De Prony found that hairdressers unemployed after the French Revolution, who were meticulous by nature, made excellent human computers. In spite of the division of labor it took eight years for the tables to be completed, and because of the inflation during the French Revolution the tables were never published in full. Portions were published for the first time in 1891.    

Though the tables remained unpublished the manuscripts could be examined and consulted. De Prony's method of production of the tables inspired Charles Babbage in the design of his Difference Engine No. 1 in 1822.

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 Hamburg] 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).

French mathematician and astronomer Pierre-Simon Laplace published Paris Traité de méchanique céleste in 5 volumes with several supplements. This work was "a treatise on celestial mechanics in the tradition of Newton’s Principia. Here Laplace applied his mathematical theories of probability to celestial bodies and concluded that the apparent changes in the motion of planets and their satellites are changes of long periods, and that the solar system is in all probability very stable. He gave methods for calculating the movements of translation and rotation of heavenly bodies and for resolving problems of tides, from which he deduced the mass of the moon” (Dibner, Heralds of Science (1980) no. 14). Laplace’s system of celestial mechanics (a term he coined) marked an advance over that of Newton, who had posited the necessity of a Deity in the universe to correct planetary irregularities; Laplace on the other hand, when asked by Napoleon why his system contained no mention of the Creator, replied “I had no need of such a hypothesis.”

The bibliographical makeup of Mécanique céleste is among the most complex of science classics; see Horblit and the Norman library catalogue for collations and paginations. Two issues of Vols. I-II exist, one with the imprint of Crapelet and Duprat alone and the French Republican date “An VII”; and one dated “1799” with the additional imprint reading “Berlin: chez F. T. de la Garde, Libraire,” printed for European distribution. The third volume contains a single separately paginated supplement (“Supplément au Traité de mécanique céleste . . . présenté au Bureau des Longitudes, le 17 août 1808”); the fourth volume has two separately paginated supplements (“Supplément au dixième livre du Traité de mécanique céleste. Sur l’action capillaire” and “Supplément à la théorie de l’action capillaire”). The fifth volume’s supplement,  (“Supplément au 5e volume du Traité de mécanique céleste . . .”) appeared in 1827. It is not unusual for sets to be lacking one or more of the supplements. Vol. V, comprising a series of addenda to the first four volumes, appeared twenty years after Vol. IV; according to Laplace’s “Avertissement” to this volume, each of its five books was issued separately in the month indicated on its part-title.

Horblit, One Hundred Books Famous in Science no. 63. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1277. Carter & Muir, Printing and the Mind of Man (1967) no. 252.

"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."

At the age of 24 Carl Friedrich Gauss published Disquisitiones arithmeticae in Leipzig, 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 published Nouvelles méthodes pour la détermination des orbites des comètes. His appendix to this work, “Sur la Méthode des moindres quarrés,” represented the first publication of the method of least squares, the earliest form of regression analysis.

Between 1806 and 1810 French astronomer and surveyer Pierre Méchain and French mathematician and astronomer Jean Delambre published Base du système mètrique décimal in 3 volumes. This work was concluded in 1821 by a fourth volume entitled Recueil d’observations géodésiques, astronomiques et physiques by French physicist, astronomer and mathematician Jean Baptiste Biot and French mathematician, physicist and astronomer François Arago.

In 1788 the French Academy of Sciences, at the suggestion of Talleyrand, proposed the establishment of a new universal decimal system of measurement founded upon some “natural and invariable base” to replace Europe’s diverse regional systems. This project was approved by the Assemblée nationale in 1790 and a basic unit or “meter (metre)” of measurement proposed, which was to be one ten-millionth of the distance between the terrestrial pole and the Equator. In 1792 Méchain and Delambre were appointed to make the necessary geodetic measurements of the meridian passing through Dunkirk and Barcelona, from which the meter would be derived, and in 1793/94 (An II of  the French Revolutionary calendar), the French government introduced the metric system to the country through the publication of Instruction sur les mesures déduites de la grandeur de la terre, uniformes pour toute la république, et sur les calculs relatifs à leur division décimale issued in Paris by the Imprimerie Nationale. 

Méchain and Delambre's scientific project was hampered by France’s political revolution, by the death of Méchain in 1804, and by the tedious calculations involved in converting one system to another; it was not until 1810 that Delambre was able to complete the final volume of the Base du système mètrique décimal.

Méchain and Delambre had determined the length of the meter by taking measurements over a meridian arc of 10 degrees. After Méchain’s death in 1804, the Bureau des Longitudes proposed that the meter’s length be redetermined more accurately by extending measurement of the arc of the meridian south to the Balearic Islands of Mallorca, Menorca and Ibiza. François Arago and Jean Baptiste Biot were assigned to this task. Arago was twenty years old at the start of this project. In 1806 he and Biot journeyed to Spain and began triangulating the Spanish coast. Their work was disrupted by the political unrest that developed after Napoleon’s invasion of Spain in 1807. Biot returned to Paris after they had determined the latitude of Formentera, the southernmost point to which they were to carry the survey. Arago continued the work until 1808, his purpose being to measure a meridian arc in order to determine the exact length of a meter.

After Biot's departure, the political ferment caused by the entrance of the French into Spain extended to the Balearic Islands, and the population suspected Arago's movements and his lighting of fires on the top of mola de l’Esclop as the activities of a spy for the invading army. Their reaction was such that he was obliged to give himself up for imprisonment in the fortress of Bellver in June 1808. On July 28 Arago escaped from the island in a fishing boat, and after an adventurous voyage he reached Algiers on August 3. From there he obtained a passage in a vessel bound for Marseille, but on August 16, just as the vessel was nearing Marseille, it fell into the hands of a Spanish corsair. With the rest the crew, Arago was taken to Roses in Catalonia, and imprisoned first in a windmill, and afterwards in a fortress, until the town fell into the hands of the French, and the prisoners were transferred to Palamós.

After three months' imprisonment, Arago and the others were released on the demand of the dey (ruler) of Algiers, and again set sail for Marseille on the November 28, but when within sight of their port they were driven back by a northerly wind to Bougie on the coast of Africa. Transport to Algiers by sea from this place would have required a delay of three months. Arago, therefore, set out over land, on what had to be a strenuous journey, guided by a Muslim imam, and reached Algiers on Christmas Day. After six months in Algiers, on June 21, 1809, Arago set sail for Marseille, where he had to undergo a monotonous and inhospitable quarantine in the lazaretto before his difficulties were over, roughly one year after he had first been imprisoned. The first letter he received, while in the lazaretto, was from Alexander von Humboldt—the origin of a scientific relationship which lasted over forty years.

In spite of the successive imprisonments, an escape, voyages, and other hardships he endured, Arago had succeeded in preserving the records of his survey; and his first act on his return home was to deposit them in the Bureau des Longitudes in Paris. As a reward for his heroic conduct in the cause of science, he was elected a member of the Académie des Sciences at the remarkably early age of twenty-three, and before the close of 1809 he was chosen by the council of the Ėcole Polytechnique to succeed Gaspard Monge in the chair of analytic geometry. At the same time he was named by the emperor one of the astronomers of the Obsérvatoire royale, which remained his residence till his death, and in this capacity he delivered his remarkably successful series of popular lectures on astronomy from 1812 to 1845. Most of Arago's later scientific contributions were in physics, particularly optics and magnetism: he discovered the phenomena of rotary magnetism (the greater sensitivity for light in the periphery of the eye) and rotary polarization, invented the first polariscope, and performed important experiments supporting the undulatory theory of light. In his capacity as secretary of the Académie des sciences, he championed the photographic process invented by Louis Daguerre, announcing its discovery to the Académie in 1839, and using his influence to obtain publicity and funding for its inventor.

Arago’s results, together with geodetic data obtained in France, England and Scotland, were published in the Recueil d’observations géodésiques, issued as a supplement to Méchain and Delambre’s work 11 years after he carried the data back to France, in 1821. Political opposition to the new system of measurement may have contributed to the unusually long delay in publication. 

Besides his scientific career Arago was a politician, representing a scientific point of view, and accomplishing government projects that were culturally valuable. For a little over one month, from May 9, 1848 to June 24, 1848 he was the 25th Prime Minister of France. Arago detailed his scientific adventures in his Histoire de ma jeunesse published the year after his death, in 1854.  This was translated into English by the Rev. Baden-Powell as History of My Youth (1855). The translation was reprinted in Arago's Biographies of Distinguished Scientific Men (1859).

As a tribute to Arago’s contribution, in 1994 the Arago Association and the city of Paris commissioned a Dutch conceptual artist, Jan Dibbets to create a memorial to Arago. Dibbets came up with the idea of setting 135 bronze Arago Medallions into the ground along the Paris Meridian between the northern and southern limits of Paris: a total distance of 9.2 kilometres/5.7 miles. Each medallion is 12 cm in diameter and marked with the name ARAGO plus N and S pointers; only 121 are documented in the official guide to the medallions. One of these was shown in the film, The Da Vinci Code.

Carter & Muir, Printing and the Mind of Man (1967) no. 260. Daumas, Arago: La jeunesse de la science, ch. IV. Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 1481.

Alder, The Measure of the World (2003) pp. 7 and 294 refers to Méchain's annotated copy of this set of books in the Karpeles Manuscript Library.  In 2011, when I finished this database entry, I owned Arago's copy of the set.

Mathematician Charles Babbage started on a model his first Difference Engine, a special-purpose machine that linked 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. In 1822 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. 

French mathematician and physicist Jean Baptiste Joseph Fourier published 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.

Mathematician and engineer Charles Babbage published "On a Method of Expressing by Signs the Action of Machinery," Philosophical Transactions 111 (1826) 250-65, 4 plates. This was the first publication of Babbage's system of mechanical notation that enabled him to describe the logic and operation of his machines on paper as they would be fabricated in metal. Babbage later stated that "Without the aid of this language I could not have invented the Analytical Engine; nor do I believe that any machinery of equal complexity can ever be contrived without the assistance of that or of some other equivalent language. The Difference Engine No. 2 . . . is entirely described by its aid" (Babbage, Passages from the Life of a Philosopher [1864], 104).  

Babbage considered his mechanical notation system to be one of his finest inventions, and thought it should be widely implemented. It was a source of frustration to him that no other machine designer adopted it (probably because no other engineer during Babbage's time attempted to build machines as logically and mechanically complex as Babbage's). More than one hundred years later, in the 1930s, when developments in logic were applied to switching systems in the earliest efforts to develop electromechanical calculators, Claude Shannon demonstrated that Boolean algebra could be applied to the same types of problems for which Babbage had designed his mechanical notation system.  

"While making designs for the Difference Engine, Babbage found great difficulty in ascertaining from ordinary drawings-plans and elevations-the state of rest or motion of individual parts as computation proceeded: that is to say in following in detail succeeding stages of a machine's action. This led him to develop a mechanical notation which provided a systematic method for labeling parts of a machine, classifying each part as fixed or moveable; a formal method for indicating the relative motions of the several parts which was easy to follow; and means for relating notations and drawings so that they might illustrate and explain each other. As the calculating engines developed the notation became a powerful but complex formal tool. Although its scope was much wider than logical systems, the mechanical notation was the most powerful formal method for describing switching systems until Boolean algebra was applied to the problem in the middle of the twentieth century. In its mature form the mechanical notation was to comprise three main components: a systematic method for preparing and labeling complex mechanical drawings; timing diagrams; and logic diagrams, which show the general flow of control" (Hyman, Charles Babbage [1982], 58).

Hook & Norman, Origins of Cyberspace (2001) no. 37.

Mathematician Nicolai Ivanovitch Lobachevskii (Lobachevsky), rector of the Kazan Imperial University, published "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. The complete work was illustrated with two folding plates. This was the first published work on non-Euclidean geometry. It appeared in the Messenger of the Kazan Imperial University 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. One copy was held in a private collection in America.

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

Mathematician János Bolyai published "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 mathematician father Farkas Bolyai, 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 (Lobachevsky), 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, Russia, 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 German mathematician 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.

Because of the destruction of most of the Maya codices in the sixteenth century, scholars had extremely limited access to the original texts. It was not until 1810 that the first reproduction of any Mayan codex— five pages from the Dresden Codex— were reproduced by Alexander von Humboldt in his Vues de cordillères, et monuments des peuples indigènes de l'Amérique. From this very limited reproduction in 1832 European-American autodidact polymath, mathematician, botanist, zoologist, and malachologist Constantine Samuel Rafinesque, while working in Philadelphia, deciphered the Maya's system of numerals.

In 1832 Rafinesque published his discovery in his periodical, the Atlantic Journal, and Friend of Knowledge: A Cyclopedic Journal and Review of Universal Science and Knowledge: Historical, Natural, and Medical Arts and Sciences: Industry, Agriculture, Education, and Every Useful Information. He announced it in a three-part article addressed to Jean-François Champollion, whose name he misspelled, "on the Graphic systems of America, and the Glyphs of Otolum or Palenque, in Central America." In the second part of this article, on page 42, Rafinesque briefly explained his discovery of the meaning of the Maya bar and dot system in which a dot equals one and a bar equals 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).

Like most of Rafinesque's numerous other publications, his Atlantic Journal enjoyed very limited success, and folded after only eight issues.  Copies of the original edition are extremely rare.  My copy is a facsimile reprint issued by the Arnold Arboretum, Boston, in 1946.

In Note sur un moyen de tracer des courbes données par des équations différentielles  French mathematician, mechanical engineer and scientist Gaspard-Gustave Coriolis described a mechanical device to integrate differential equations of the first order. This was the beginning of researches on solution of differential equations using mechanical devices.

American writer, poet, editor, literary critic, and magazinist Edgar Allan Poe published in the Southern Literary Messenger issued from Richmond, Virginia "Maelzel's Chess Player." In this article on automata Poe provided a very closely reasoned explanation of the concealed human operation of von Kempelen's Turk, which Poe had seen exhibited in Richmond by Maelzel a few weeks earlier. 

Poe also briefly compared von Kempelen's Turk to Babbage's Difference Engine No. 1, which was limited to the computation of astronomical and navigation tables, suggesting essentially that if the Turk was fully automated and had the ability to use the results of one logical operation to make a decision about the next one—what was later called "conditional branching" —it would be far superior to Babbage's machine.  This feature was, of course, later designed into Babbage's Analytical Engine. 

Here is Poe's comparison of the two machines:

"But if these machines were ingenious, what shall we think of the calculating machine of Mr. Babbage? What shall we think of an engine of wood and metal which can not only compute astronomical and navigation tables to any given extent, but render the exactitude of its operations mathematically certain through its power of correcting its possible errors? What shall we think of a machine which can not only accomplish all this, but actually print off its elaborate results, when obtained, without the slightest intervention of the intellect of man? It will, perhaps, be said, in reply, that a machine such as we have described is altogether above comparison with the Chess-Player of Maelzel. By no means — it is altogether beneath it — that is to say provided we assume (what should never for a moment be assumed) that the Chess-Player is a pure machine, and performs its operations without any immediate human agency. Arithmetical or algebraical calculations are, from their very nature, fixed and determinate. Certain data being given, certain results necessarily and inevitably follow. These results have dependence upon nothing, and are influenced by nothing but the data originally given. And the question to be solved proceeds, or should proceed, to its final determination, by a succession of unerring steps liable to no change, and subject to no modification. This being the case, we can without difficulty conceive the possibility of so arranging a piece of mechanism, that upon starting it in accordance with the data of the question to be solved, it should continue its movements regularly, progressively, and undeviatingly towards the required solution, since these movements, however complex, are never imagined to be otherwise than finite and determinate. But the case is widely different with the Chess-Player. With him there is no determinate progression. No one move in chess necessarily follows upon any one other. From no particular disposition of the men at one period of a game can we predicate their disposition at a different period. Let us place the first move in a game of chess, in juxta-position with the data of an algebraical question, and their great difference will be immediately perceived. From the latter — from the data — the second step of the question, dependent thereupon, inevitably follows. It is modelled by the data. It must be thus and not otherwise. But from the first move in the game of chess no especial second move follows of necessity. In the algebraical question, as it proceeds towards solution, the certainty of its operations remains altogether unimpaired. The second step having been a consequence of the data, the [column 2:] third step is equally a consequence of the second, the fourth of the third, the fifth of the fourth, and so on, and not possibly otherwise, to the end. But in proportion to the progress made in a game of chess, is the uncertainty of each ensuing move. A few moves having been made, no step is certain. Different spectators of the game would advise different moves. All is then dependent upon the variable judgment of the players. Now even granting (what should not be granted) that the movements of the Automaton Chess-Player were in themselves determinate, they would be necessarily interrupted and disarranged by the indeterminate will of his antagonist. There is then no analogy whatever between the operations of the Chess-Player, and those of the calculating machine of Mr. Babbage, and if we choose to call the former a pure machine we must be prepared to admit that it is, beyond all comparison, the most wonderful of the inventions of mankind. Its original projector, however, Baron Kempelen, had no scruple in declaring it to be a "very ordinary piece of mechanism — a bagatelle whose effects appeared so marvellous only from the boldness of the conception, and the fortunate choice of the methods adopted for promoting the illusion." But it is needless to dwell upon this point. It is quite certain that the operations of the Automaton are regulated by mind, and by nothing else. Indeed this matter is susceptible of a mathematical demonstration, a priori. The only question then is of the manner in which human agency is brought to bear. Before entering upon this subject it would be as well to give a brief history and description of the Chess-Player for the benefit of such of our readers as may never have had an opportunity of witnessing Mr. Maelzel's exhibition."

Belgian mathematician Pierre François Verhulst published from Brussels "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).

Italian mathematician and politician Luigi Federico Menabrea published "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 (Turin) Italy 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, translated Menabrea’s paper, "Notions sur la machine analytique de M. Charles Babbage" (1842).

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.)

Augusta Ada, Countess of Lovelace, composed a letter to Charles Babbage concerning her notes to Menabrea's paper on programming Babbage's planned Analytical Engine. This autograph letter, preserved in the British Library (Add. MS 37192 folios 362v-363), includes the following text:

"I want to put in something about Bernouilli's Numbers, in one of my Notes, as an example of how an implicit function may be worked out by the engine, without  having been worked out by human head & hands first. Give me the necessary data and formulae."

In January 2011 an image of this letter was available on the British Library's cell phone app called Treasures. 

The letter is notable for confirming that Ada's knowledge of mathematics was limited, and that she may have mainly contributed poetic language to her annotations of the English translation of Menabrea's key paper, while incorporating mathematical examples written by Babbage.

Because of Ada's fame as Byron's daughter, and her social position as the Countess of Lovelace, Babbage hoped that Ada's translation and annotation of Menabrea's paper would help promote building the Analytical Engine, a mechanical general purpose programmable computer that he conceived and designed roughly one hundred years ahead of its time.

English mathematician and philosopher George Boole published a pamphlet entitled The Mathematical Analysis of Logic — his first exposition of 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.

In 1847 mathematician, logician and pioneer collector of the history of mathematics, Augustus de Morgan published 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 the first separately published bibliography on the history of science excluding economics; McCulloch's annotated bibliography of the history of economics preceded it in 1845. The bulk of de Morgan's 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. De Morgan stated that he was able to acquire his library 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.”

After de Morgan's death in 1871 his library of about 4500 books, pamphlets, manuscripts and autograph letters was purchased by British banker and politician Samuel Jones-Loyd, 1st Baron Overstone and donated by him to the University of London, becoming the first special collection at the Unversity of London library. Even though de Morgan’s library was not kept together when it was transferred, his books were separately identified in the printed catalogue of the Library published in 1876. Thus it is possible to study one of the pioneering collections of books formed in England not just on mathematics, but on a wide range of the history of physical sciences. In 2012 the Senate House Library of the University of London showed examples from de Morgan's library on its website: http://www.ull.ac.uk/specialcollections/demorganexploration.shtml

In his book, The Process of Thought Adapted to Words and Language, English surgeon Alfred Smee suggested the possibility of information storage and retrieval by a mechanical logical machine operating analogously to the human mind.

This was an attempt to produce an artificial system of reasoning based upon neurological principles which were then primarily a matter of speculation. The problem was that Smee's hypothetical “electro-biological” machine, built out of mechanical parts, which he conceived in generality but had no way of engineering, or building even in part, might have occupied a space larger than London.

In 1851 Dar al-Funun (Persian: دارالفنون‎), the first modern institution of higher learning in Persia, was established in Tehran. Conceived as a polytechnic to train upper-class Persian youth in Medicine, Engineering, Military Science, and Geology, Dar-al-Funun was founded by Amir Kabir, then the royal vizier to Nasereddin Shah, the Shah of Iran.  "It was similar in scope and purpose to American land grant colleges like Purdue and Texas A&M. Like them, it developed and expanded its mission over the next hundred years, eventually becoming the University of Tehran" (Wikipedia article on Dar al-Funun, accessed 05-24-2012).

In 1854 English mathematician and philosopher George Boole published 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, published his paper on Mental Calculation. (See Reading 3.1)

In 1857 Jules Antoine Lissajous, professor of mathematics at the Lycée Saint-Louis in Paris, published "Mémoire sur l'Etude optique des mouvements vibratoires,"  Annales de chimie et de physique, 3rd series, 51 (1857) 147-232, 2 folding plates. Lissajous's paper on his optical method of studying vibration gave rise to the widely used "Lissajous figures," or Lissajous curves, defined mathematically as curves in the xy plane generated by the functions y = a sin (w1t + q1) and x = b sin (w2t + q2) where w1 and w2 are small integers.

"Like some other physicists of his time, Lissajous was interested in demonstrations of vibration that did not depend on the sense of hearing. . . . [His] most important research, first described in 1855, was the invention of a way to study acoustic vibrations by reflecting a light beam from the vibrating object onto a screen. . . . Lissajous produced two kinds of luminous curves. In the first kind, light is reflected from a tuning fork (to which a small mirror is attached), and then from a large mirror that is rotated rapidly. . . . The second kind of curve, named the 'Lissajous figure,' is more useful. The light beam is successively reflected from mirrors on two forks that are vibrating about mutually perpendicular axes. Persistence of vision causes various curves, whose shapes depend on the relative frequency, phase, and amplitude of the forks' vibrations. . . . If one of the forks is a standard, the form of the curve enables an estimate of the parameters of the other. As Lissajous said, they enable one to study beats (the ellipses rotate as the phase difference changes). 'Lissajous figures' have been, and still are, important in this respect" (DSB).

Lissajous figures are sometimes used in graphic design as logos. Examples include the logos of the Australian Broadcasting Corporation (a = 1, b = 3, d = p/2) and the Lincoln Laboratory at MIT (a = 4, b = 3, d = 0).

Prior to modern computer graphics, Lissajous curves were typically generated using an oscilloscope. Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure. Lissajous curves can also be traced mechanically by means of a harmonograph. They often appear in computer screensavers.

English mathematician, engineer and computer designer Charles Babbage published his autobiography, Passages from the Life of a Philosopher, in which he presented the most detailed descriptions of his Difference and Analytical Engines published during his lifetime, and wrote 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 published "A Dynamical Theory of the Electro-Magnetic Field" in the Transactions of the Royal Society of London.  In this culminating paper on the foundations of electromagnetic theory Maxwell developed twenty field equations of electromagnetism, clinching the theory that light was a form of electricity.
Maxwell had already found in 1862 a link of a purely phenomenological kind between electromagnetic quantities and the velocity of light, but this 1865 paper provided a new theoretical framework for the subject, based on experiment and a few general dynamical principles, from which the propagation of electromagnetic waves through space followed without special assumptions about molecular vortices or forces between electrical particles. The paper provided a theoretical framework, based on experiment and a few general dynamical principles, for the propagation of electromagnetic waves through space.

English mathematician and economist William Stanley Jevons constructed his “logical piano,” the first logic machine to solve complicated problems with superhuman speed.  Jevons described his machine in "On the mechanical performance of logical inference," Philosophical Transactions of the Royal Society 160 (1870), 497-518 with 3 plates.

First demonstrated before the Royal Society in 1870, the original logical piano is still on display in the Museum of the History of Science, Oxford. The internal structure of the machine is illustrated in the three plates accompanying Jevons' paper, which provide a reasonable guide to its construction. Jevons was a pioneer of symbolic logic, and his paper includes a detailed explanation of his system of equational logic, which derived from (and in some important ways improved) the symbolic logic devised by Boole over two decades earlier.

Gardner, Logic Machines and Diagrams, 91-103. Schabas, A World Ruled by Number, 54ff. Lee, Computer Pioneers, 400-401. Randell, The Origins of Digital Computers,  479.

Belgian astronomer, mathematician, statistician and sociologist Lambert Adolphe Jacques Quetelet published in Brussels 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.

Charles Babbage’s scientific library was sold at auction in 1872. The auction catalogue, containing over two thousand items on topics such as mathematical tables, cryptography, and calculating machines, and including many rare volumes, may be the first catalogue of a library on computing and its history.

American theoretical physicist, physical chemist, and mathematician Josiah Willard Gibbs published "On the Equilibrium of Heterogeneous Substances," Transactions of the Connecticut Academy of Arts and Sciences III, 108-248; 343-524. Gibbs’s paper, known as the Principia of chemical thermodynamics and physical chemistry, remains, along with Benjamin Franklin’s pioneering studies of electricity, among the greatest American contributions to physics and chemistry. The long two-part paper integrated chemical, physical, electrical, and electromagnetic phenomena into a coherent system. It introduced concepts such as chemical potential, phase rule, and others which form the basis for modern physical chemistry.

“In this monumental, densely woven, 300-page treatise, the first law of thermodynamics, the second law of thermodynamics, the fundamental thermodynamic relation, are applied to the predication and quantification of thermodynamic reaction tendencies in any thermodynamic system in a visual, three-dimensional graphical language of Lagrangian calculus and phase changes, among others” (Wikipedia article On the Equilibrium of Heterogeneous Substances, accessed 06-23-2011).

Though Gibbs’s work was published in one of the most obscure of American scientific periodicals, Gibbs attempted to gain wider circulation for his ideas by mailing a larger than usual number of offprints of the papers to scientists he believed would be interested. One of the few scientists who read the first offprint he received and commented about it in print was James Clerk Maxwell, “On the Equilibrium of Heterogeneous Substances,” Proceedings of the Cambridge Philosophical Society II (1876), 427-30. However, it is unclear that the papers had wide influence in the scientific community until they were translated into German by Wilhelm Ostwald (1892) and into French by Henry Louis Le Chatelier (1899). Through these translations and later editions Gibbs’s work influenced numerous scientists, including Nobel laureates in chemistry, physics, and economics, as expounded by the Wikipedia:

◊ Johan van der Waals of the Netherlands won the 1910 Nobel prize in physics. In his Nobel Lecture, he acknowledged the influence on his work of Gibbs's equations of state.

◊ "Max Planck of Germany won the 1918 Nobel prize in physics for his work in quantum mechanics, particularly his 1900 quantum theory paper. This work is largely based on the thermodynamics of Rudolf Clausius, Gibbs, and Ludwig Boltzmann. Nevertheless, Planck said about Gibbs: "…whose name not only in America but in the whole world will ever be reckoned among the most renowned theoretical physicists of all times."

◊ "At the turn of the 20th century, Gilbert N. Lewis and Merle Randall used and extended Gibbs's work on chemical thermodynamics, published their results in the 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, one of the two founding books in chemical thermodynamics. In the 1910s, William Giauque entered the College of Chemistry at Berkeley, where he received a bachelor of science degree in chemistry, with honors, in 1920. At first he wanted to become a chemical engineer, but soon developed an interest in chemical research under Lewis's influence. In 1934, Giauque became a full Professor of Chemistry at Berkeley. In 1949, he won the Nobel Prize in Chemistry for his studies in the properties of matter at temperatures close to absolute zero, studies guided by the third law of thermodynamics.

◊ "Gibbs strongly influenced the education of the economist Irving Fisher, who was awarded the first Yale Ph.D. in economics in 1891. One of Gibbs's protegés was Edwin Bidwell Wilson, who in turn passed his Gibbsian knowledge to the American economist Paul Samuelson. In 1947, Samuelson published Foundations of Economic Analysis, based on his Harvard University doctoral dissertation. Samuelson explicitly acknowledged the influence of the classical thermodynamic methods of Gibbs. Samuelson was the sole recipient of the Nobel Prize in Economics in 1970, the second year of the Prize. In 2003, Samuelson described Gibbs as "Yale's great physicist" (Wikipedia article on Josiah Willard Gibbs, accessed 06-23-2011).

Remarkably Gibbs’s mailing lists for distribution of his offprints are among his papers preserved at Yale. These lists were published by Wheeler, Josiah Willard Gibbs. The History of a Great Mind (1952) 235-48. According to these records Gibbs mailed nearly 100 copies of each of the two parts of his paper, mostly to individuals, and 10 each to institutions. Of these few appear to have survived. Dibner, Heralds of Science  no. 49 (journal issue). Horblit, One Hundred Books Famous in Science  no. 60 (journal issue). Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 899 (offprint issue).

About 1875 engineer Frank S. Baldwin of Philadelphia and Willgot Theophil Odhner, a Swedish engineer and entrepreneur working in St. Petersburg, Russia, independently invented calculators using a true variable-toothed gear. This was the first real advance in mechanical calculating technology since Gottfried Leibniz's stepped drum (1673). These calculators were called "pinwheel calculators."

The greater ease of use of this technology, its general reliability, and the compact size of the equipment incorporating it caused an explosion of sales in the calculator industry.

Swedish inventor Martin Wiberg used 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 was the first logarithmic table produced by a calculating machine. The device is preserved at Tekniska museet (The Technical Museum) of Sweden in Stockholm. 

Bruno Abdank-Abakanowicz, a mathematician, inventor and electrical engineer, invented 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).

Abdank-Abakanowicz published a monograph entitled Les Intégraphes (Paris, 1886).

Friedrich Ludwig Gottlob Frege published 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 of the City of London School  Edwin A. Abbott published a work of scientific fantasy or mathematical fiction 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, were finally published in an abbreviated form in one volume. They were the most monumental work of calculation ever carried out by human computers.

France. Service Geographique de l'Armee. Tables des logarithmes a huit decimales des nombres entiers de 1 a 120000 et des sinus et tangentes de dix secondes en dix secondes d'arc dans le systeme de la division centesimale du quadrant. Paris: Imprimerie Nationale, 1891.

Hook & Norman, Origins of Cyberspace (2001) no. 301.

French engineer and applied mathematician Philbert Maurice d'Ocagne published 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 Scientific Biography X [1974] 170). 

The recently established Deutsche Mathematiker-Vereinigung held 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.

On January 14, 1898 the Reverend Charles Lutwidge Dodgson, the English author, mathematician, logician, Anglican clergyman, and photographer, best known by his pen name, Lewis Carroll, died. He had spent nearly his entire life at Christ Church College, Oxford, in various capacities. In addition to his published writings, which included Alice in Wonderland, Dodgson maintained a meticulous ledger recording his incoming and outgoing correspondence over his lifetime. As a reflection of how many letters an individual could exchange in this era before telephone, Dodgson/Carroll wrote or received approximately 98,000 letters.

In 1900, at the beginning of a new century, German mathematician and physicist David Hilbert of the University of Göttingen published in Mathematische Probleme a list of twenty-three problems that he predicted would 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 published 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 Library, 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.

From 1910 to 1913 British philosopher, logician, mathematician, historian, and social critic Bertrand Russell and English mathematician and philosopher Alfred North Whitehead published Principia mathematica in three volumes, taking up the task — first attempted in Russell's never completed 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 published "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).

The Napier Tercentenary Celebration  marking the three hundredth anniversary of the publication of Napier's Mirifici logarithmorum canonis descriptio (1614), was held at the Royal Society of Edinburgh from July 24 to July 27, 1914 — just five days before the start of World War I. Participants in the exhibition included individuals and companies from Scotland, England, France, and Germany. The meeting was intended to include a colloquium on the mathematics of computation, but that was canceled because war was considered imminent.

A celebration of Napier's pivotal role in the history of calculation, the exhibition featured displays of many different types of calculating machines, as well as exhibits of other aids to calculation such as mathematical tables, the abacus and slide rules, planimeters and other integrating devices, and ruled papers and nomograms. These were described in the Napier Tercentenary Celebration. Handbook to the Exhibition, which contained separate sections, with chapters by various contributors, devoted to each type of calculating device. Among the notable chapters is Percy E. Ludgate's "Automatic Calculating Machines" (pp. 124-27): apart from Ludgate's "On a proposed analytical machine" (Scientific Proceedings of the Royal Dublin Society 12 [1909]: 77-91), this chapter contains the only discussion of his improvements to Babbage's Analytical Engine (none of which was ever realized). Also of note is W. G. Smith's "Notes on the Special Development of Calculating Ability" (pp. 60-68), discussing human "lightning calculators" and mathematically gifted "idiot savants," such as were employed by Gauss. Prior to the advent of electronic digital computers, these human computers were often faster than their mechanical counterparts.

The most widely used tools for calculation at the time of the Napier tercentenary were mathematical tables, which are thoroughly surveyed, explained, and described in the Handbook (bibliographical descriptions of the rare mathematical tables exhibited were published the following year in the Napier Tercentenary Memorial Volume. The Handbook also contains a large illustrated section on calculating machines, which were divided into four types: (1) stepped-gear machines based on the Leibnitz wheel, such as those of Charles Xavier Thomas de Colmar; (2) machines with variable-toothed gears, such as the Brunsviga; (3) key-set machines like those made by Burroughs; and (4) key-driven machines such as those made by Felt and Tarrant.

The Handbook was published in two forms: a softcover version presented to those who registered for the exhibition; and a hardcover version issued for sale under the title Modern Instruments and Methods of Calculation. Relatively few copies of the softcover version seem to have been distributed at the exhibition, partly because the exhibition took place in Edinburgh, but mainly because war broke out just after it began. Most copies were bound in cloth and sold in London.

"The events of the First World War caused no less upheaval in the world of computing than in the rest of society. A great many technical changes, such as the ever-increasing use of punched-card accounting machines, were to cause computing to assume a different character in the time between the two World Wars. Thus the Handbook should be viewed as a report on the state of the art just before these changes were to begin taking place" (Williams 1982, [x]).  

Hook & Norman, Origins of Cyberspace (2001) no. 322.

German mathematician Leopold Löwenheim of Berlin published 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, professor at Technische Universität Wien, introduced the Radon transform. He also demonstrated 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.

In 1920 Norwegian mathematician Thoraf Albert Skolem of the University of Oslo proved 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.

Italian Italian mathematician and physicist, known for his contributions to mathematical biology Vito Volterra of Sapienza – Università di Roma published "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.

Astronomer and mechanical computation pioneer Leslie J. Comrie working in London discovered how to use a commercial accounting machine as a difference engine.

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

In 1998 information theorist Ralph V. R. Hartley of Bell Labs published “Transmission of Information,” in which he proved "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.

Hungarian-American mathematician, physicist, economist and polymath John von Neumann then working at Humboldt-Universität zu Berlin, published "Zur Theorie der Gesellschaftsspiele" in Mathematische Annalen, 100, 295–300. This paper "On the Theory of Parlor Games" propounded the minimax theorem, inventing the theory of games.

In 1930 French mathematician Jacques Herbrand, a student at the École Normale Supérieure in Paris, published his doctoral thesis, Recherches sur la théorie de la démonstration. It was 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.

In 1931 Austrian logician, mathematician and philosopher Kurt Gödel published while in Vienna Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "On Formally Undecidable Propositions of 'Principia Mathematica' and Related Systems"). That article dated November 17, 1930, which first appeared in the 1931 volume of Monatshefte für Mathematik, contained Godel's first and second incompleteness theorems. 

van Heijenoort, ed. From Frege to Gödel: A Source Book on Mathematical Logic 1879–1931 (1967) 592-617.

Engineer Vannevar Bush began 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.

American mathematician and logician Alonzo Church of Princeton published his logical proof of the undecidability of arithmetic, using his lambda calculus.

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

English mathematician Alan Turing published On Computable Numbers, a mathematical description of what he called a universal machine— an astraction that could, in principle, solve any mathematical problem that could 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.)

Mathematician and logician Alonzo Church of Princeton called Alan Turing’s universal machine a Turing Machine. (See Reading 7.2.)

Independently of Alan Turing, mathematician and logician Emil Post of the City College of New York developed a mathematical model of computation that was essentially equivalent to the Turing machine. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometime's called "Post's machine" or a Post-Turing machine."

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

George Stibitz, a research mathematician at Bell Telephone Labs in New York City, built 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 completed 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.

English mathematician, logician, cryptanalyst, and computer scientist Alan Turing reported to the Government Code and Cypher School, Bletchley Park, in the town of Bletchley, England.

Having collaborated with engineer Julian Bigelow, mathematician Norbert Wiener published, as a classified document from MIT, 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, accepted a position at MIT to work with Norbert Wiener.

Mathematical Tables and Other Aids to Computation (MTAC), the world’s first computing journal, began publication in Washington, D.C.

At this time mathematical tables prepared by human computers were the primary calculating aid. The journal reported on the new electromechanical and electronic “aids to computation” as they were developed.

Mathematician, physicist, and economist John von Neumann and economist Oskar Morgenstern published The Theory of Games and Economic Behavior in Princeton at the University Press.

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.

Faced with mathematical computations regarding the Atomic bomb that were too time-consuming for human computers, mathematician and physicist John von Neumann visited the ENIAC two-accumulator system for the first time, and became deeply interested in the project.

Mathematician and physicist John von Neumann of Princeton  privately circulated copies of his First Draft on a Report on the EDVAC to twenty-four people connected with the EDVAC project. This document, written between February and June 1945, provided the first theoretical description of the basic details of a stored-program computer: what later became known as the Von Neumann architecture.

To avoid the government's security classification, and to avoid engineering problems that might detract from the logical considerations under discussion, Von Neumann avoided mentioning specific hardware. Influenced by Alan Turing and by Warren McCulloch and Walter Pitts, von Neumann patterned the machine to some degree after human thought processes. (See Reading 8.1.)

In June 2009 I was able to download a PDF of the text of von Neumann's report at this link: http://www.virtualtravelog.net/entries/2003-08-TheFirstDraft.pdf.

Howard Aiken of Harvard University published 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.

Mathematician Max Newman founded the computer laboratory at Manchester University via a grant from the Royal Society.

A contest was 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 combined 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).

In 1948 mathematician Norbert Wiener at MIT published Cybernetics or Control and Communication in the Animal and the Machine, a widely circulated and influential book that applied theories of information and communication to both biological systems and machines. Computer-related words with the “cyber” prefix, including "cyberspace," originate from Wiener’s book. Cybernetics was also the first conventionally published book to discuss electronic digital computing. Writing as a mathematician rather than an engineer, Wiener’s discussion was theoretical rather than specific. Strangely the first edition of the book was published in English in Paris at the press of Hermann et Cie. The first American edition was printed offset from the French sheets and issued by John Wiley in New York, also in 1948. I have never seen an edition printed or published in England. 

Independently of Claude Shannon, Wiener conceived of communications engineering as a brand of statistical physics and applied this viewpoint to the concept of information. Wiener's chapter on "Time series, information, and communication" contained the first publication of Wiener's formula describing the probability density of continuous information. This was remarkably close to Shannon's formula dealing with discrete time published in A Mathematical Theory of Communication (1948). Cybernetics also contained a chapter on "Computing machines and the nervous system." This was a theoretical discussion, influenced by McCulloch and Pitts, of differences and similarities between information processing in the electronic computer and the human brain. It contained a discussion of the difference between human memory and the different computer memories then available. Tacked on at the end of Cybernetics were speculations by Wiener about building a chess-playing computer, predating Shannon's first paper on the topic.

Cybernetics is a peculiar, rambling blend of popular and highly technical writing, ranging from history to philosophy, to mathematics, to information and communication theory, to computer science, and to biology. Reflecting the amazingly wide range of the author's interests, it represented an interdisciplinary approach to information systems both in biology and machines. It influenced a generation of scientists working in a wide range of disciplines. In it were the roots of various elements of computer science, which by the mid-1950s had broken off from cybernetics to form their own specialties. Among these separate disciplines were information theory, computer learning, and artificial intelligence.

It is probable that Wiley had Hermann et Cie supervise the typesetting because they specialized in books on mathematics.  Hermann printed the first edition by letterpress; the American edition was printed offset from the French sheets. Perhaps because the typesetting was done in France Wiener did not have the opportunity to read proofs carefully, as the first edition contained many typographical errors which were repeated in the American edition, and which remained uncorrected through the various printings of the American edition until a second edition was finally published by John Wiley and MIT Press in 1961. 

Though the book contained a lot of technical mathematics, and was not written for a popular audience, the first American edition went through at least 5 printings during 1948,  and several later printings, most of which were probably not read in their entirety by purchasers. Sales of Wiener's book were helped by reviews in wide circulation journals such as the review in TIME Magazine on December 27, 1948, entitled "In Man's Image." The reviewer used the word calculator to describe the machines; at this time the word computer was reserved for humans.

"Some modern calculators 'remember' by means of electrical impulses circulating for long periods around closed circuits. One kind of human memory is believed to depend on a similar system: groups of neurons connected in rings. The memory impulses go round & round and are called upon when needed. Some calculators use 'scanning' as in television. So does the brain. In place of the beam of electrons which scans a television tube, many physiologists believe, the brain has 'alpha waves': electrical surges, ten per second, which question the circulating memories.

"By copying the human brain, says Professor Wiener, man is learning how to build better calculating machines. And the more he learns about calculators, the better he understands the brain. The cyberneticists are like explorers pushing into a new country and finding that nature, by constructing the human brain, pioneered there before them.

"Psychotic Calculators. If calculators are like human brains, do they ever go insane? Indeed they do, says Professor Wiener. Certain forms of insanity in the brain are believed to be caused by circulating memories which have got out of hand. Memory impulses (of worry or fear) go round & round, refusing to be suppressed. They invade other neuron circuits and eventually occupy so much nerve tissue that the brain, absorbed in its worry, can think of nothing else.

"The more complicated calculating machines, says Professor Wiener, do this too. An electrical impulse, instead of going to its proper destination and quieting down dutifully, starts circulating lawlessly. It invades distant parts of the mechanism and sets the whole mass of electronic neurons moving in wild oscillations" (http://www.time.com/time/magazine/article/0,9171,886484-2,00.html, accessed 03-05-2009).

Presumably the commercial success of Cybernetics encouraged Wiley to publish Berkeley's Giant Brains, or Machines that Think in 1949.

♦ In October 2012 I offered for sale the copy of the first American printing of Cybernetics that Wiener inscribed to Jerry Wiesner, the head of the laboratory at MIT where Wiener conducted his research. This was the first inscribed copy of the first edition (either the French or American first) that I had ever seen on the market, though the occasional signed copy of the American edition did turn up. Having read our catalogue description of that item, my colleague Arthur Freeman emailed me this story pertinent to Wiener's habit of not inscribing books:

"Norbert, whom I grew up nearby (he visited our converted barn in Belmont, Mass., constantly to play frantic theoretical blackboard math with my father, an economist/statistician at MIT, which my mother, herself a bit better at pure math, would have to explain to him later), was a notorious cheapskate. His wife once persuaded him to invite some colleagues out for a beer at the Oxford Grill in Harvard Square, which he did, and after a fifteen-minute sipping session, he got up to go, and solemnly collected one dime each from each of his guests. So when *Cybernetics* appeared on the shelves of the Harvard Coop Bookstore, my father was surprised and flattered that Norbert wanted him to have an inscribed copy, and together they went to Coop, where Norbert duly picked one out, wrote in it, and carried it to the check-out counter--where he ceremoniously handed it over to my father to pay for. This was a great topic of family folklore. I wonder if Jerry Wiesner paid for his copy too?"

Mathematician John von Neumann delivered lectures at the University of Illinois at Urbana-Champaign on The Theory of Self-Reproducing Automata. In these lectures von Neumann showed that in theory a program could reproduce itself. The lectures were completed and edited by A. W. Burks and published by the University of Illinois Press 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 was turned off for the last time at the Aberdeen Proving Ground.

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 did not finish his last book, The Computer and the Brain, in which he compared the functions of computers and the human brain.

The first published use of the term "software" in a computing context is often credited to American statistician John W. Tukey, who published the term in "The Teaching of Concrete Mathematics," American Mathematical Monthly, January 9, 1958. Tukey wrote:

"Today the 'software' comprising the carefully planned interpretive routines, compilers, and other aspects of automative programming are at least as important to the modern electronic calculator as its 'hardware' of tubes, transistors, wires, tapes and the like" (http://www.maa.org/mathland/mathtrek_7_31_00.html, accessed 02-02-2010).

Note that Tukey referred to computers as "calculators." Up to this time the word "computer" typically referred to people, and the use of the word computer for a machine was just coming into popular use.

However, the priority of Tukey in this context appears to be unjustified. On April 30, 2013 Paul Niquette informed me that Richard R. Carhart used the term in the  Proceedings of the Second National Symposium on Quality Control and Reliability in Electronics: Washington, D.C., January 9-10, 1956. It is, of course, possible – even likely – that others used the word in spoken, rather than printed, context before either Carhart or Tukey. Niquette states that he used the term as early as 1953.

At the 1956 Dartmouth summer session on artificial intelligence, Allen Newell and Herbert Simon demonstrated the first AI program, the Logic Theorist, to find the basic equations of logic as defined in Principia Mathematica by Whitehead and Russell.

For one of the equations, the Logic Theorist surpassed its inventors’ expectations by finding a new and better proof. This was the “the first foray by artificial intelligence research into high-order intellectual processes” (Feigenbaum and Feldman, Computers and Thought [1963]).

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

Allan Newell, Clifford Shaw, and Herbert Simon invented “game tree pruning,” an artificial intelligence technique.

Having been computed by human computers since 1767, the Nautical Almanac was 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, in 1960 Mathematical Tables and Other Aids to Computation (MTAC), the first computing journal or periodical, published by the American Mathematical Society, changed its name to Mathematics of Computation.

Philosopher, mathematician and computer scientist John Alan Robinson, while at Rice University, published "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.

American mathematician James W. Cooley of IBM Watson Research Center, Yorktown Heights, New York,  and American statistician John W. Tukey published "An algorithm for the machine calculation of complex Fourier series", Mathematics of  Computation 19, 297–301. This paper enunciated the Cooley-Tukey FFT algorithm, the most common fast Fourier transform algorithm.

"The motivation for it [FFT algorithm] was provided by Dr. Richard L. Garwin at IBM Watson Research who was concerned about verifying a Nuclear arms treaty with the Soviet Union for the SALT talks. Garwin thought that if he had a very much faster Fourier Transform he could plant sensors in the ground in countries surrounding the Soviet Union. He suggested the idea of how Fourier transforms could be programmed to be much faster to both Cooley and Tukey. They did the work, the sensors were planted, and he was able to locate nuclear explosions to within 15 kilometers of where they were occurring" (Wikipedia article on James Cooley, accessed 03-06-2012).

Texas Instruments filed the patent for the first hand-held electronic calculator, invented by Jack S. Kilby, Jerry Merryman, and Jim Van Tassel. The patent (Number 3,819,921) was awarded on June 25, 1974.

This miniature calculator employed a large-scale integrated semiconductor array containing the equivalent of thousands of discrete semiconductor devices.

While a professor at UCLA, Italian-American electrical engineer and businessman Andrew Viterbi developed 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).

Hewlett Packard, Palo Alto, California, introduced the programmable desk calculator, the HP 9100A.

 "HP called it a desktop calculator, because, as Bill Hewlett said, 'If we had called it a computer, it would have been rejected by our customers' computer gurus because it didn't look like an IBM. We therefore decided to call it a calculator, and all such nonsense disappeared.' An engineering triumph at the time, the logic circuit was produced without any integrated circuits; the assembly of the CPU having been entirely executed in discrete components. With CRT display, magnetic-card storage, and printer, the price was around $5000. The machine's keyboard was a cross between that of a scientific calculator and an adding machine. There was no alphabetic keyboard" (Wikipedia article on Hewlett-Packard, accessed 03-10-2010).

In 1975 French American mathematician, physicist, economist, and information theorist Benoit Mandelbrot, a researcher at the IBM T. J. Watson Research Center, Yorktown Heights, New York, first developed 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 expanded and translated his ideas in his book Fractals: Form, Chance and Dimension (1977). He further expanded them in The Fractal Geometry of Nature (1982). In 1999 American Scientist magazine stated that these three books, taken together, comprise “one of the ten most influential scientific essays of the 20th century.” The impact of these books on the scientific community, and on the educated public, was significantly enhanced by mathematically accurate computer-drawn illustrations created by programmers working with Mandelbrot, primarily at IBM Research. Images for the 1977 and 1982 books were mainly by Richard F. Voss. The early graphics were low-resolution black and white; later drawings were higher resolution and in color as computer graphic technology evolved between 1975 and 1982.

Mandelbrot's new geometry made it possible to describe mathematically the kinds of irregularities existing in nature, and had applications in an enormously wide range of scientific and technological fields.

British physicist Peter Mansfield developed a mathematical technique that would allow NMR scans to take seconds rather than hours and produce clearer images than the technique  Paul Lauterbur developed in 1973.

Mansfield showed how gradients in the magnetic field could be mathematically analysed, which made it possible to develop a useful nuclear magnetic resonance imaging technique. Mansfield also showed how extremely fast imaging could be achievable. This became technically possible a decade later.

P Mansfield and A A Maudsley, Medical imaging by NMR, Brit. J. Radiol. 50 (1977) 188.
P Mansfield, Multi-planar imaging formation using NMR spin echoes J. Physics C. Solid State Phys. 10 (1977) L55–L58.

The references from Mansfield's Nobel Lecture. You can also watch a 64 minute video of Mansfield delivering his lecture at this link.

Between 1977 and 1979 computer scientist Donald E. Knuth of Stanford University created the TeX page-formatting language and the Metafont character shape specification language, originally as a way of improving the typography of his own publications. These he described in four publications in 1979:

1. "Mathematical Typography," Bulletin (New Series) of the American Mathematical Society, March 1979, Vol. 1, No. 2, 337-72. Josiah Willard Gibbs Lecture, January 4, 1978.

2. TEX, a system for technical text.  A manual published by the American Mathematical Society, June, 1979.

3. Metafont, a system for alphabet design, September, 1979.

4. In December 1979 Digital Press in Bedford, Massachusetts, a division of Digital Equipment Corporation (DEC), together with the American Mathematical Society issued these three documents in book form as TEX and Metafont. New Directions in Typesetting, with a Foreward by C. Gordon Bell, then Vice President of Engineering at DEC, and a Preface by Knuth.

Preceding the development and wide-acceptance of PostScript (1984) and TrueType (1991), expectations for the impact of TeX and Metafont were appropriately great within the computer community. As a reflection of this, I quote Gordon Bell's 1979 introduction in full:

"Don Knuth's Tau Epison Chi (TeX) is potentially the most significant invention in typesetting in this century. It introduces a standard language for computer typography and in terms of importance could rank near the introduction of the Gutenberg press. The TeX system:

"•understands typography from individual charcters to page design;

"•permits any typewriter, word processing system, computer-based editor, or TeX system editor to be used as an input device with a standard language;

"•can typeset various formats and languages;

"•is structured to be user-extendable to virtually all applications.

"These improvements are benchmarks in typesetting and text creation. To date, computer-based typesetting systems have simply facilitated typesetting. Moreover, the proliferation of word processing systems makes possible the widespread direct transmission of text to typesetting without the intervening typesetting process—provided we use the standard language that TeX offers.

"A direct link between text input and typesetting will permit a drastic restructuring of the journal- and book-publishing industry, allowing it to be oriented substantially more toward the author. Unitl now, even authors with word processing equipment have been unable to participate in the representation of their message in print. Prior to Gutenberg's invention, manuscripts were conceived and designed simultaneously, and often the author's hand shaped the entire final product. The results were beautiful and varied, in contrast to the manufacture of most modern books, which vary only in cover design. With TeX, moreover not only can the author influence his own format and representation, but he also can produce more accurate material than can be rapidly mass-produced, shortening the time between idea and dissemination.

"TeX is significant as a standard language because of the way it understands typography using a framework of boxes and glue in a hierarchical fashion so that any font, page layout, or other typesetting parameter can be set. This is in striking contrast to most typesetting systems, which are built with no generality. Finally, the input form is user-defined by means of a macroprocessor so that virutally any text can be input and can control the typography part of the program. It is this generality and segmentation of function that makes TeX significant.

"This book is about much more than just the Tex system. The Gibbs Lecture presents the twin themes of how typography can help mathematics and how mathematics can help typography, and the material on METAFONT is intriguing and useful in its description of the use of mathematics in type design.

"While the emphasis of TeX is on mathematics, the system is equally applicable to and will no doubt be used in many other domains. Don Knuth, in fact, shows us precisely how the system can humanize basic communciations.

"At Digital, we hope to use TeX immediately, I urge others to adopt and use it so that the language standard can be established.

My copy of the first printing of TeX and Metafont was presented to the San Francisco book designer and book historian Adrian Wilson in February, 1980. Wilson worked in both letterpress and offset and designed many prize-winning books. On the first page of Bell's Foreward Wilson made pencil notes in the margin, taking issue with three points in the third paragraph. It is not clear that Wilson read past the Foreward; however, the points that Wilson made remain valid:

1. "Prior to Gutenberg's invention, manuscripts were conceived and designed simultaneously, and often the author's hand shaped the entire final product." Here Wilson commented, "Very rarely!"  I am unaware of any manuscripts prior to printing, except perhaps for author's manuscripts or the extremely few autograph manuscripts that survived, where it can be demonstrated that the author "shaped the final product" in the sense of its physical appearance on the page rather than in the textual sense. In addition, the process of manuscript copying by different scribes tended to make each manuscript copy different in subtle, or not so subtle ways, from each other.

2. "The results were beautiful and varied, in contrast to the manufacture of most modern books, which vary only in cover design." Here Wilson commented, "not so."  Bell's statement ignored, of course, the incredible diversity of all aspects of the design of "modern books" in addition to their covers.

3. "With TeX, moreover, not only can the author influence his own format and representation. . . ." Here Wilson commented, "author as designer! no." Before desktop publishing (1984-85) the ability of authors who were not programmers to design an acceptable looking book was, of course, highly limited. Even in 2012, when I wrote this database entry, few authors without expert knowledge of book design or graphic arts expertise could produce a genuinely attractively designed book.

Knuth continued his typographic work, issuing a second and larger volume entitled Digital Typography in 1999. This contains a remarkable collection of stories and technical papers concerning the continuation of his work in typography. In 2012 TeX and Metafont remained niche products for composing and scientific books and papers with the market dominated by PostScript and TrueType. As Richard Southall commented in Printer's type in the twentieth century. Manufacturing and design methods (2005) 224, footnote 6, "Donald Knuth's Metafont language, with its radically different approach to the specification of character image configurations, might have provided an alternative, and many ways a better, approach to typemaking if the interface it presented to designers had not been so forbidding."

On March 12, 2013 at a meeting of the Colophon Club in Berkeley, California I heard Knuth deliver a fascinating presentation on how and why he developed TeX and Metafont.  From this I gathered more general understanding of Knuth's system, which from the very beginning he placed in the public domain, and from which he never intended to profit. A more technical explanation of why TeX and Metafont remained niche products may be found in this posting from the Typophile.com website on December 15, 2004: 

"Metafont can only produce bitmap fonts which is a severe limitation. Nowadays, people usually create outline fonts since they are scalable and usable in different resolutions. There are tools that convert .mf to Type 1 or TrueType but this is done by autotracing which results in rather poor quality.

"There is a related product called Metapost, created by John Hobby, which allows parametric creation of PostScript graphics. This was later extended by Boguslaw Jackowski, Piotr Strzelczyk and Janusz Nowacki to MetaType1, an outline-based parametric font creation system. However, just like many other parametric font creation systems (e.g. Font Chameleon, Infinifont, LiveType), it never gained the necessary momentum. With no professional support and no solid user interface, the tools for creating these sorts of fonts were never able to reach a broad user base. Even Multiple Master fonts that had good user interface tools (Fontographer, FontLab) were dropped because handling them turned out to be too complicated and the revenues were too limited.

"Developing mature applications is a long and laborous effort. The commercial market is difficult, which is visible with the fact that numerous efforts such as Fontographer, FontStudio , TypeDesigner or RoboFog "died". The open source community is too weak to develop a good specialty tool of that sort (open source projects work well with mass products such as Mozilla or OpenOffice, with hundreds of engineers working in their spare time or on government/organizational funding).

"Today, with the exception of DTL FontMaster and FontForge (which is free), FontLab is the only font creation application that is actively being developed. First version of FontLab was created 12 years ago and in that process, we have learned that a good user interface is crucial to a success.

"Font creators are mostly designers, not engineers. They need visual tools. Also, type is often too subtle to rely on parametric creation. While it would be tempting to re-use the exactly same shape of a serif on n, m, i and l, often, subtle changes need to be made for best effect. The more subtle and refined the letterforms get, the less the parametric approach is useful. Donald Knuth's Computer Modern isn't a particularly well-designed typeface and frankly, I have never seen a good typeface made with Metafont.

"When people make a profession out of creating type, i.e. they make their living on type design, the issue of a tool being free becomes less relevant. Also, tools such as Metafont are only nominally free. There are no licensing costs but there are substantial costs of maintenance, support and learning. The learning curves are steep, the user communities are small and not integrated, there is no professional support. Therefore, if you work with tools such as Metafont, you're often left on your own. This is a fact often overlooked by those who advertise free or open source software.

"There is a good selection of links about parametric font creation at: http://www.myfonts.com/activity/parametric-fonts/" (accessed 03-13-2013).

At Lehigh University, Bethlehem, Pennsylvania, Frederick Cohen demonstrated a virus-like program on a VAX11/750 system. The program was 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, Champaign, Illinois, introduced 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).

Peter F. Brown and colleagues at IBM's Thomas J. Watson Research Center, Yorktown Heights, NY, published "The Mathematics of Statistical Machine Translation: Parameter Estimation," Computational Linguistics, 19 (2) 263-311:

"We describe a series of five statistical models of the translation process and give algorithms for estimating the parameters of these models given a set of pairs of sentences that are translations of one another. We define a concept of word-by-word alignment between such pairs of sentences. For any given pair of such sentences each of our models assigns a probability to each of the possible word-by-word alignments. We give an algorithm for seeking the most probable of these alignments. Although the algorithm is suboptimal, the alignment thus obtained accounts well for the word-by-word relationships in the pair of sentences. We have a great deal of data in French and English from the proceedings of the Canadian Parliament. Accordingly, we have restricted our work to these two languages; but we,feel that because our algorithms have minimal linguistic content they would work well on other pairs of languages. We also feel, again because of the minimal linguistic content of our algorithms, that it is reasonable to argue that word-by-word alignments are inherent in any sufficiently large bilingual corpus."

"The first ideas of statistical machine translation were introduced by Warren Weaver in 1949, including the ideas of applying Claude Shannon's information theory. Statistical machine translation was re-introduced in 1991 by researchers at IBM's Thomas J. Watson Research Center and has contributed to the significant resurgence in interest in machine translation in recent years. Nowadays it is by far the most widely-studied machine translation method" (Wikipedia article on Statistical machine translation, accessed 05-14-2010).

Anthropologists Gary Urton and Carrie Brezine published "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, a theoretical physicist and computational epidemiologist at Northwestern University in Evanston, Illinois, L. Hufnagel, and T. Geisel published "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 reported 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. After a decade of scientific study all the Archimedes Palimpsest images were released to the public on Google Books on October 29, 2008. At the time this was the earliest text available on Google Books. 

♦ Several videos, audio presentations and articles about the project are available at www.archimedespalimpsest.org

Petr Sojka of the Department of Computer Graphics and Design of Faculty of Informatics, Masaryk University, Czech Republic, organized 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])?"

A highly interdisciplinary group of scientists, primarily from Harvard University: Jean-Baptiste Michel,Yuan Kui Shen, Aviva P. Aiden, Adrian Veres, Matthew K. Gray, The Google Books Team, Joseph P. Pickett, Dale Hoiberg, Dan Clancy, Peter Norvig, Jon Orwant, Steven Pinker, Martin A. Nowak and Erez Lieberman Aiden published "Quantitative Analysis of Culture Using Millions of Digitized Books," Science, Published Online December 16 2010 Science 14 January 2011: Vol. 331 no. 6014 pp. 176-182 DOI: 10.1126/science.1199644

The authors were associated with the following organizations: Program for Evolutionary Dynamics, Institute for Quantitative Social Sciences Department of Psychology, Department of Systems Biology Computer Science and Artificial Intelligence Laboratory, Harvard Medical School, Harvard College Google, Inc. Houghton Mifflin Harcourt Encyclopaedia Britannica, Inc. Department of Organismic and Evolutionary Biology Department of Mathematics, Broad Institute of Harvard and MITCambridge School of Engineering and Applied Sciences Harvard Society of Fellows, Laboratory-at-Large.

This paper from the Cultural Observatory at Harvard and collaborators represented the first major publication resulting from The Google Labs N-gram (Ngram) Viewer,

"the first tool of its kind, capable of precisely and rapidly quantifying cultural trends based on massive quantities of data. It is a gateway to culturomics! The browser is designed to enable you to examine the frequency of words (banana) or phrases ('United States of America') in books over time. You'll be searching through over 5.2 million books: ~4% of all books ever published" (http://www.culturomics.org/Resources/A-users-guide-to-culturomics, accessed 12-19-2010).

"We constructed a corpus of digitized texts containing about 4% of all books ever printed. Analysis of this corpus enables us to investigate cultural trends quantitatively. We survey the vast terrain of "culturomics", focusing on linguistic and cultural phenomena that were reflected in the English language between 1800 and 2000. We show how this approach can provide insights about fields as diverse as lexicography, the evolution of grammar, collective memory, the adoption of technology, the pursuit of fame, censorship, and historical epidemiology. "Culturomics" extends the boundaries of rigorous quantitative inquiry to a wide array of new phenomena spanning the social sciences and the humanities" (http://www.sciencemag.org/content/early/2010/12/15/science.1199644, accessed 12-19-2010).  

"The Cultural Observatory at Harvard is working to enable the quantitative study of human culture across societies and across centuries. We do this in three ways: Creating massive datasets relevant to human culture Using these datasets to power wholly new types of analysis Developing tools that enable researchers and the general public to query the data" (http://www.culturomics.org/cultural-observatory-at-harvard, accessed 12-19-2010).

"Today the data we have available to make predictions has grown almost unimaginably large: it represents 2.5 quintillion bytes of data each day, Mr. Silver tells us, enough zeros and ones to fill a billion books of 10 million pages each. Our ability to tease the signal from the noise has not grown nearly as fast. As a result, we have plenty of data but lack the ability to extract truth from it and to build models that accurately predict the future that data portends" ("Mining Truth From Data Babel. Nate Silver’s ‘Signal and the Noise’ Examines Predictions"  By Leonard Mlodinow, NYTimes.com 10-23-2012).