# The Mayan Mathematics

The Mayan civilization is generally dated from 1500 B.C.E to 1700 C.E. although the so-called Classic Period stretches from about 250 CE to 900 CE. At its peak, it was one of the most densely populated and culturally dynamic societies in the world. The Yucatan Peninsula (see map below) in Mexico was the scene for the development of one of the most advanced civilizations of the ancient world. The Mayans had a sophisticated ritual system that was overseen by a priestly class. This class of priests developed a philosophy with time as divine and eternal.32 The calendar, and calculations related to it, were thus very important to the ritual life of the priestly class, and hence the Mayan people. In fact, much of what we know about this culture comes from their calendar records and astronomy data. Another important source of information on the Mayans is the writings of Father Diego de Landa, who went to Mexico as a missionary in 1549.

The Maya World.

Hernán Cortés, excited by stories of the lands which Columbus had recently discovered, sailed from Spain in 1505 landing in Hispaniola which is now Santo Domingo. After farming there for some years he sailed with Velázquez to conquer Cuba in 1511. He was twice elected major of Santiago then, on 18 February 1519, he sailed for the coast of Yucatán with a force of 11 ships, 508 soldiers, 100 sailors, and 16 horses. He landed at Tabasco on the northern coast of the Yucatán peninsular. He met with little resistance from the local population and they presented him with presents including twenty girls. He married Malinche, one of these girls.

The people of the Yucatán peninsular were descendants of the ancient Mayan civilisation which had been in decline from about 900 AD. It is the mathematical achievements of this civilisation which we are concerned with in this article. However, before describing these, we should note that Cortés went on to conquer the Aztec peoples of Mexico. He captured Tenochtitlán before the end of 1519 (the city was rebuilt as Mexico City in 1521) and the Aztec empire fell to Cortés before the end of 1521. Malinche, who acted as interpreter for Cortés, played an important role in his ventures.

In order to understand how knowledge of the Mayan people has reached us we must consider another Spanish character in this story, namely Diego de Landa. He joined the Franciscan Order in 1541 when about 17 years old and requested that he be sent to the New World as a missionary. Landa helped the Mayan peoples in the Yucatán peninsular and generally tried his best to protect them from their new Spanish masters. He visited the ruins of the great cities of the Mayan civilisation and learnt from the people about their customs and history.

However, despite being sympathetic to the Mayan people, Landa abhorred their religious practices. To the devote Christian that Landa was, the Mayan religion with its icons and the Mayan texts written in hieroglyphics appeared like the work of the devil. He ordered all Mayan idols be destroyed and all Mayan books be burned. Landa seems to have been surprised at the distress this caused the Mayans.

Nobody can quite understand Landa's feelings but perhaps he regretted his actions or perhaps he tried to justify them. Certainly what he then did was to write a book Relación de las cosas de Yucatán (1566) which describes the hieroglyphics, customs, temples, religious practices and history of the Mayans which his own actions had done so much to eradicate. The book was lost for many years but rediscovered in Madrid three hundred years later in 1869.

A small number of Mayan documents survived destruction by Landa. The most important are: the Dresden Codex now kept in the Sächsische Landesbibliothek Dresden; the Madrid Codex now kept in the American Museum in Madrid; and the Paris Codex now in the Bibliothèque nationale in Paris. The Dresden Codex is a treatise on astronomy, thought to have been copied in the eleventh century AD from an original document dating from the seventh or eighth centuries AD.

The Dresden codex:

Knowledge of the Mayan civilisation has been greatly increased in the last thirty years (P T Culbert and J A Sabloff, Maya civilisation New York, 1995 and .J A Sabloff, The new archaeology and the ancient Maya, London, 1990.) Modern techniques such as high resolution radar images, aerial photography and satellite images have changed conceptions of the Maya civilisation. We are interested in the Classic Period of the Maya which spans the period 250 AD to 900 AD, but this classic period was built on top of a civilisation which had lived in the region from about 2000 BC.

The Maya of the Classic Period built large cities, around fifteen have been identified in the Yucatán peninsular, with recent estimates of the population of the city of Tikal in the Southern Lowlands being around 50000 at its peak. Tikal is probably the largest of the cities and recent studies have identified about 3000 separate constructions including temples, palaces, shrines, wood and thatch houses, terraces, causeways, plazas and huge reservoirs for storing rainwater. The rulers were astronomer priests who lived in the cities who controlled the people with their religious instructions. Farming with sophisticated raised fields and irrigation systems provided the food to support the population.

A common culture, calendar, and mythology held the civilisation together and astronomy played an important part in the religion which underlay the whole life of the people. Of course astronomy and calendar calculations require mathematics and indeed the Maya constructed a very sophisticated number system. We do not know the date of these mathematical achievements but it seems certain that when the system was devised it contained features which were more advanced than any other in the world at the time.

# The Maya Number System

There were two numeral systems developed by the Mayans _ one for the common people and one for the priests. Not only did these two systems use different symbols, they also used different base systems. For the priests, the number system was governed by ritual. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads, 33 like the sample to the left 34. Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. This makes for a confusing system, the details of which we will skip.

In order to write numbers down, there were only three symbols needed in this system. A horizontal bar represented the quantity 5, a dot represented the quantity 1, and a special symbol (thought to be a shell) represented zero. The Mayan system may have been the first to make use of zero as a placeholder/number. The first 20 numbers are shown in the table to the right.

The decimal mathematical system widely used today originated by counting with the fingers a person has. Counting with the fingers and toes started the Maya vigesimal system. So it is based on groups of twenty units. Just as the decimal system goes by 1, 10, 100, 1000, 10000, etc., the Maya vigesimal system goes 1, 20, 400, 8000, 160000, etc. While in the decimal system there are ten possible digits for each placeholder [0 - 9], in the Maya vigesimal system each placeholder has a possible twenty digits [0 - 19]. For example, in the decimal system 31 = 10 * 3 + 1 while in the vigesimal system 31 = 20 + 11. The Maya discovered and used the zero. Their zero is represented by an ovular shell.

Unlike our system, where the ones place starts on the right and then moves to the left, the Mayan systems places the ones on the bottom of a vertical orientation and moves up as the place value increases.

When numbers are written in vertical form, there should never be more than four dots in a single place. When writing Mayan numbers, every group of five dots becomes one bar. Also, there should never be more than three bars in a single place, four bars would be converted to one dot in the next place up. It?s the same as 10 getting converted to a 1 in the next place up when we carry during addition.

The Maya number system was a base twenty system.

Here are the Mayan numerals.

Mayan Numerals.

# Characteristics of The Maya Mathematical System:

a) It is vigesimal, this means that it is based on 20 units [0 - 19] instead of the 10 units [0 - 9] of the decimal system.

b) It only uses three symbols, alone or combined, to write any number. These are: the dot - worth 1 unit, the bar - worth 5 units and the zero simbolized by a shell.

c) It also uses a vigesimal positioning system, in which numbers in higher places grow multiplied by 20?s instead of the 10's of the decimal system, compare number 168,421 in both systems:

 Place Number 168,421 Place's   Decimal value Equals & is written Place's Vigesimal value Equals & is written 6th 1 X 100,000= 100,000 3'200,000 5th 6 X 10,000 = 60,000 160,000 4th 8 X 1,000 = 8,000 8,000 3rd 4 X 100 = 400 400 2nd 2 X 10 = 20 20 1st 1 X 1 = 1 1 TOTAL Arabic 168,421 Maya 168,421

d) Numbers in the Maya system can be written vertically or horizontally. In vertical writing, the bars are placed horizontally and the dots go on top of them, in this case the vigesimal positions grow up from the base. When written horizontally, the bars are placed vertically and the dots go to their left and higher vigesimal positions grow to the left of the first entry

Mayan Names For Numbers

 0 xix im 10 lahun 1 hun 11 buluc 20 hun kal 400 hun bak 2 caa 12 lahca 40 ca kal 800 ca bak 3 ox 13 oxlahun 60 ox kal 1200 ox bak 4 can 14 canlahun 80 can kal 1600 can bak 5 hoo 15 hoolahun 100 hoo kal 2000 hoo bak 6 uac 16 uaclahun 120 uac kal 8,000 pic 7 uuc 17 uuclahun 140 uuc kal 160,000 calab 8 uaxac 18 uaxaclahun 200 ka hoo kal 3'200,000 kinchil 9 bolon 19 bolonlahun 300 ox hoo kal 64,000,000 alau

Thus when writing vertically the vigesimal positioning system, to write 20 a zero is placed in the first position (base) with a dot on top of it, in the second position. The dot in this place means one unit of the second order which equals to 20. To write 21, the zero would change to a dot (1 unit) and for the subsequent numbers the original 19 number count will follow in the first position. As they in turn reach 19 again another unit (dot) is added to the second position. Any number higher than 19 units in the second position is written using units of the third position. A unit of the third position is worth 400 (20 x 20), so to write 401 a dot goes in the first position, a zero in the second and a dot in the third. Positions higher than the third also grow multiplied by twenties from the previous ones. Examples of the numbers mentioned above follow:

Mathematical Count.

(Note : the Maya made one exception to this order, only in their calendric calculations they gave the third position a value of 360 instead of 400, the higher positions though, are also multiplied by 20.)

Calendric Count.

Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes. Although it was a base 20 system, called a vigesimal system, one can see how five plays a major role, again clearly relating to five fingers and toes. In fact it is worth noting that although the system is base 20 it only has three number symbols (perhaps the unit symbol arising from a pebble and the line symbol from a stick used in counting). Often people say how impossible it would be to have a number system to a large base since it would involve remembering so many special symbols. This shows how people are conditioned by the system they use and can only see variants of the number system in close analogy with the one with which they are familiar. Surprising and advanced features of the Mayan number system are the zero, denoted by a shell for reasons we cannot explain, and the positional nature of the system. However, the system was not a truly positional system as we shall now explain.

In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20's up to 19, the next the number of 400's up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20's up to 19, it changes in the third place and this denotes the number of 360's up to 19 instead of the number of 400's. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 202, the next the number of 18 × 203 and so on. For example [ 8;14;3;1;12 ] represents

12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203 = 1253912.
As a second example [ 9;8;9;13;0 ] represents
0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 =1357100.
Both these examples are found in the ruins of Mayan towns and we shall explain their significance below.

Now the system we have just described is used in the Dresden Codex and it is the only system for which we have any written evidence. In (G Ifrah, A universal history of numbers : From prehistory to the invention of the computer, London, 1998.) Ifrah argues that the number system we have just introduced was the system of the Mayan priests and astronomers which they used for astronomical and calendar calculations. This is undoubtedly the case and that it was used in this way explains some of the irregularities in the system as we shall see below. It was the system used for calendars. However Ifrah also argues for a second truly base 20 system which would have been used by the merchants and was the number system which would also have been used in speech. This, he claims had a circle or dot (coming from a cocoa bean currency according to some, or a pebble used for counting according to others) as its unity, a horizontal bar for 5 and special symbols for 20, 400, 8000 etc. Ifrah writes (G Ifrah, A universal history of numbers : From prehistory to the invention of the computer London, 1998.):-

Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.
Let us say a little about the Maya calendar before returning to their number systems, for the calendar was behind the structure of the number system. Of course, there was also an influence in the other direction, and the base of the number system 20 played a major role in the structure of the calendar.

The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 "months" of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short "month" of only 5 days that was called the Wayeb. The Wayeb was considered an unlucky period and Landa wrote in his classic text that the Maya did not wash, comb their hair or do any hard work during these five days. Anyone born during these days would have bad luck and remain poor and unhappy all their lives.

Why then was the ritual calendar based on 260 days? This is a question to which we have no satisfactory answer. One suggestion is that since the Maya lived in the tropics the sun was directly overhead twice every year. Perhaps they measured 260 days and 105 days as the successive periods between the sun being directly overhead (the fact that this is true for the Yucatán peninsular cannot be taken to prove this theory). A second theory is that the Maya had 13 gods of the "upper world", and 20 was the number of a man, so giving each god a 20 day month gave a ritual calendar of 260 days.

At any rate having two calendars, one with 260 days and the other with 365 days, meant that the two would calendars would return to the same cycle after lcm(260, 365) = 18980 days. Now this is after 52 civil years (or 73 ritual years) and indeed the Maya had a sacred cycle consisting of 52 years. Another major player in the calendar was the planet Venus. The Mayan astronomers calculated its synodic period (after which it has returned to the same position) as 584 days. Now after only two of the 52 years cycles Venus will have made 65 revolutions and also be back to the same position. This remarkable coincidence would have meant great celebrations by the Maya every 104 years.

Now there was a third way that the Mayan people had of measuring time which was not strictly a calendar. It was an absolute timescale which was based on a creation date and time was measured forward from this. What date was the Mayan creation date? The date most often taken is 12 August 3113 BC but we should say straightaway that not all historians agree that this was the zero of this so-called "Long Count". Now one might expect that this measurement of time would either give the number of ritual calendar years since creation or the number of civil calendar years since creation. However it does neither.

The Long Count is based on a year of 360 days, or perhaps it is more accurate to say that it is just a count of days with then numbers represented in the Mayan number system. Now we see the probable reason for the departure of the number system from a true base 20 system. It was so that the system approximately represented years. Many inscriptions are found in the Mayan towns which give the date of erection in terms of this long count. Consider the two examples of Mayan numbers given above. The first

[ 8;14;3;1;12 ]
is the date given on a plate which came from the town of Tikal. It translates to
12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203
which is 1253912 days from the creation date of 12 August 3113 BC so the plate was carved in 320 AD.

The second example

[ 9;8;9;13;0 ]
is the completion date on a building in Palenque in Tabasco, near the landing site of Cortés. It translates to
0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203
which is 1357100 days from the creation date of 12 August 3113 BC so the building was completed in 603 AD.

We should note some properties (or more strictly non-properties) of the Mayan number system. The Mayans appear to have had no concept of a fraction but, as we shall see below, they were still able to make remarkably accurate astronomical measurements. Also since the Mayan numbers were not a true positional base 20 system, it fails to have the nice mathematical properties that we expect of a positional system. For example

[ 9;8;9;13;0 ] = 0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 = 1357100
yet
[ 9;8;9;13 ] = 13 + 9 × 20 + 8 × 18 × 20 + 9 × 18 × 202 = 67873.
Moving all the numbers one place left would multiply the number by 20 in a true base 20 positional system yet 20 × 67873 = 1357460 which is not equal to 1357100. For when we multiple [ 9;8;9;13 ] by 20 we get 9 × 400 where in [ 9;8;9;13;0 ] we have 9 × 360.

We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers. Yet the Mayan number system is certainly capable of being used for the operations of multiplication and division as the authors of (J B Lambert, B Ownbey-McLaughlin, and C D McLaughlin, Maya arithmetic, Amer. Sci. 68 (3) (1980), 249-255.) demonstrate.

## The Maya Astronomy

Finally we should say a little about the Mayan advances in astronomy. Rodriguez writes in (L F Rodriguez, Astronomy among the Mayans (Spanish), Rev. Mexicana Astronom. Astrofis. 10 (1985), 443-453. ):-
The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.
The Mayans carried out astronomical measurements with remarkable accuracy yet they had no instruments other than sticks. They used two sticks in the form of a cross, viewing astronomical objects through the right angle formed by the sticks. The Caracol building in Chichén Itza is thought by many to be a Mayan observatory. Many of the windows of the building are positioned to line up with significant lines of sight such as that of the setting sun on the spring equinox of 21 March and also certain lines of sight relating to the moon.

The Caracol building in Chichén Itza:

With such crude instruments the Maya were able to calculate the length of the year to be 365.242 days (the modern value is 365.242198 days). Two further remarkable calculations are of the length of the lunar month. At Copán (now on the border between Honduras and Guatemala) the Mayan astronomers found that 149 lunar months lasted 4400 days. This gives 29.5302 days as the length of the lunar month. At Palenque in Tabasco they calculated that 81 lunar months lasted 2392 days. This gives 29.5308 days as the length of the lunar month. The modern value is 29.53059 days. Was this not a remarkable achievement?

There are, however, very few other mathematical achievements of the Maya. Groemer (H Groemer, The symmetries of frieze ornaments in Maya architecture, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 203, 1994, 101-116.) describes seven types of frieze ornaments occurring on Mayan buildings from the period 600 AD to 900 AD in the Puuc region of the Yucatán. This area includes the ruins at Kabah and Labna. Groemer gives twenty-five illustrations of friezes which show Mayan inventiveness and geometric intuition in such architectural decorations.

## The Medieval Mathematics

During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as "How many angels can stand on the point of a needle?"

Roman Abacus.

From the $4th$ to $12th$ Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius? translations of some of the works of ancient Greek masters such as Nicomachus and Euclid. All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models.

By the $12th$ Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi's important book on algebra into Latin in the 12th Century, and the complete text of Euclid's "Elements" was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm.

The advent of the printing press in the $mid-15th$ Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education.

Europe's first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so'called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the $13th$ Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series $1?1 + 1?2 + 1?3 + 1?4 + 1?5...$ is a divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book "De Triangulis", in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print.

Oresme Graph.

Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor . He also held some distinctly non-standard intuitive ideas about the universe and the Earth's position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler.

## The Renassence Period

The Renaissance period was not only a new era of Humanism, but also a revival of Platonism in which mathematics was the key for understanding the universe. This belief was manifested by Kepler's model of the solar system and Vincenzo Galilei's twelve-tone equal temperament.

Kepler Model

The era of the Renaissance in Europe has been viewed as a critical turning point in western culture since it inherited the doctrine of Scholasticism, which is more likely a God-centered thought, and initiated a comprehensive study of Humanism, treating humankind?s value as the first priority. Following ancient Greek philosopher Protagoras' dogma that "man is the measure of all things", humanists sought to explore the relationships and mediated conflicts among the universe, religion, and humans. It is full of such mixed characteristics that it is a fascinating topic in historical study.

Twelve-tone equal temperament, Vincenzo_Galilei.

Scholasticism in the Middle Ages has often been viewed as a conservative and trite thought. While exploring the origins of innovative spirit of Renaissance culture, however, Durand (1943) not only considered Scholasticism as an inner tradition causing the intellectual mutation of the Renaissance, but also asserted that scholastic interpretations of Aristotelianism constitutes the fundamental part of philosophy and sciences in the $15th$ century. Scholasticism emphasized logical relationships between reason and faith. Peter Abelard? (1079 - 1142), an early master of Scholasticism, claimed that "doubt is the road to inquiry" and "by inquiry we perceive the truth" (cited in Dampier, 1966, p. 80). His attempts at revealing potential connections between truths and religious apocalypses via dialectical thinking became a paradigm of following scholastic thoughts. The most influential scholastic thinker, Thomas Aquina (1225-1274), indicated there were two valid sources for knowledge. One is the theology advocated by the church, and the other is truths derived by logical reasoning. As Aquina saw it, the two sources may not be in confronting positions. Rather, they play complimentary roles for revealing apocalypses from God. This doctrine establishes a belief that nature is a system with regular patterns in which every event and object are intimately connected by a universal law. Nonetheless, it is also such a belief causing the decline of Scholasticism. As Damper (1966) put it, "Scholasticism had trained them to destroy itself" (p. 96).

## The Italian Abacists

The Italian abacists of the 14th century were instrumental in teaching the merchants the "new" Hindu-Arabic decimal place-value system and the algorithms for using it. There was formidable resistance to this system, in Italy and most of Europe. These abacists had thoroughly studied Arabic mathematics, which emphasized algebraic methods. In fact, for many years Roman numerals were used to keep account ledgers. The old system of counting boards required the board plus a bag of counters. The new system required only pen and paper. By and by, as with new technologies in general, the superior Hindu-Arabic system won out. This took centuries

Italian Abacists.

## New Algebraic Techniques.

Unlike Islamic algebra, which was entirely rhetorical, the abacists allowed the use of symbols for unknowns. Standard words were:

 Italian English cosa thing censo square cubo cube radice root p̄ piáu (plus) m̄ meno (minus)

From Antonio de' Mazzinhi (1353-1383), known for his cleverness in solving algebraic problems, we have the example. " Find two numbers such that multiplying one by the other makes 8 and the sum of their squares is 27 "

The solution begins by supposing that the first number is un cosa meno la radice d'alchuna 'quantita' (a thing minus the root of some quantity) while the second number equals una cosa piu la radice d'alchuna 'quantita' (another thing plus the root of some quantity). We have

$$(x - \sqrt{y})(x + \sqrt{y}) = 8$$ $$(x - \sqrt{y})^2 + (x + \sqrt{y})^2 = 27$$ And the answer is $x= \frac{\sqrt{43}}{2}$, $y= \frac{11}{4}$.

## Higher Degree Equations

Another innovation of the abacists was their extention of the Islamic quadratic solving techniques to higher order equations. Of course, each text began with the standard six type of quadratics as described by Al-Khwarizmi. But many went further.

Maestro Dardi of Pisa in a 1344 work extended this list to 198 types of equations of degree up to four, some involving radicals. He gave an example of how to solve a particular cubic equation, but the methods would not generalize. Another mathematician of this age was Luca Pacioli (1445-1517), who was renown for his teaching, and his library with volumes collected over many years. He also collaborated with Leonardo da Vinci.

## Renaissance Mathematicians

German Nikolaus Von Cusa, (Latin Nicolaus Cusanus), mathematician, scholar, experimental scientist, and influential philosopher stressed the incomplete nature of man's knowledge of God and of the universe. He was ordained in 1440, became a cardinal in Brixon and in 1450 was elevated to bishop there. He was interested in geometry and logic. He contributed to the study of infinity, studying the infinitely large and the infinitely small. He also studied astronomy producing a book. For Nicholas, God and only God is absolutely infinite. The universe reflects this divine perfection by being relatively infinite, with no circumference and no center.

Cusa is best known as a philosopher on the incomplete nature of mankind's knowledge of the universe. He regarded the circle as the limit of regular polygons which he then applied in his religious teachings to demonstrate the nearness of yet the unattainability of truth. In other words, he claimed that the search for truth was equal to the task of squaring the circle. This second statement reveals the continuing and profound influence of the ancient Greeks on mathematicians and even upon the church itself.

Among Cusa's other interests were diagnostic medicine and applied science. He emphasized knowledge through experimentation, a break with Peripatetic views. He anticipated the work of the astronomer Copernicus by suggesting a movement in the universe not entirely geocentric in nature. In his study of plant growth, he concluded that plants absorb nourishment from the air. He also produced a map of Europe, and founded a hospital at his birthplace in 1458.

Johann Mueller.

Johann Muller Regiomontanus (1436 - 1476) Regiomontanus was born Johann Muller of K ® onigsberg ® but he used the Latin version of his name (Konigsberg = îKing's ® mountainî). Regiomontanus. At the age of fourteen he entered the University of Vienna. He became enchanted with all things mathematical and astronomical. Vowing to read Ptolemy's Almagest in is original, Muller studied Greek ® in Italy and devoured all available texts, whether in Greek or Latin, whether on astronomy or mathematics. Upon his return to Vienna, he taught his new learning achieving phenomenal success. He was called to Nuremberg, where a sponsor built him a full scale observatory. His instruments were the finest, whether purchased or improved by himself. In a remarkable quote taken from a letter to another mathematician, we feel his total joy in mathematics and astronomy.

Nicholas Chuquet.

Nicolas Chuquet, (c. 1445 -c. 1500), a French physician, Chuquet wrote Triparty en la science des nombres (1484), a work on algebra and arithmetic in three parts. However, it was not printed until 1880. The mathematics in Triparty was known to the Islamic algebraists, but Triparty is the first detailed algebra in French. Below are a few details on the nature and style of this work

Luca Pacioli.

Luca Pacioli (1445 - 1517), an Italian, published a Latin translation of Euclid's Elements , the first printed edition. He left unpublished a work on recreational problems, geometrical problems and proverbs. It makes frequent reference to Leonardo da Vinci who worked with him on the project.

Pacioli was a teacher of some renown. He also assembled a vast collection of mathematical materials over some 20 years. Pacioli wrote Summa de arithmetica, geometrica proportioni et proportionalita (1494) It gives a summary of the mathematics known at that time.

Summa studies arithmetic, algebra, geometry and trigonometry and provided a basis for the major progress in mathematics which took place in Europe shortly after this time. It overshadowed Chuquet's Triparty but was not mathematically significant. Subsequently, Chuquet's work was scarcely mentioned.

## The Solution of the cubic

Only a few years have now past since Andrew Wiles solved the famed Fermat's Last Theorem.4 This problem, posed in the $17th$ century, was solved in the $20th$ century. In about $1540$ the first analytical solution of a general cubic equation was determined. Mathematicians had been working toward the solution for at least $1500$ years. Solving the cubic was one of the significant feats of the next few centuries that would unshackle the European mathematicians from their Greek heritage.

The solution is richly intertwined with human endeavor not usually associated with mathematics. The cast of characters was singularly unsavory, with lies and genius co-mingled, with public disputes and contests, and with broken promises.

Among the most famous are: Geronimo Cardano (1501-1576) Ars magna (1545)

Niccolo Tartaglia (ca. 1500-1577)

Ludovico Ferrari (1522-1565)

Scipio del Ferro (ca 1465-1526)

Antonio Maria Fiore ($ist$ and $2nd$ century), student

Annibale della Nave (1500-1558), student

Girolamo_Cardano.

Girolamo Cardano (1501 - 1576) is famed for his work Ars Magna which was the first Latin treatise devoted solely to algebra. Girolamo Cardano's name was Cardan in Latin and in English he is sometimes known as Jerome Cardan.

Cardano's life was anything but conventional. In his professions, and there were several, his output was voluminous. He wrote 230 books. Of those 138 were printed. Others he burned. Among his works, he discussed painting and color in De subtilitate rerum (1551) and physical knowledge of the day in De rerum varietate (1557).

One of his last works was his autobiography, De vita propria liber (A Book of My Own Life), is as singularly remarkable as a biography as Ars Magna is in algebra. Published when he was seventy four, he analyzes and confesses with startling candor his habits, character, mind, likes and dislikes, virtues and vices, honors, errors, illnesses, eccentricities, and dreams. He charges himself with obstinacy, bitterness, pugnacity, cheating at gambling, and vengefulness. He lists failures, particularly the proper rearing of his sons. A physician, he discusses his numerous, often surprising cures. He also reveals a great number of disabilities, including sexual disfunction, stuttering, palpitation, colic, dysentery, hemorrhoids, gout, and more. This was one of the very first modern autobiographies. Although we know him for his mathematics, his achievements

Cardano studied at Pavia and Padua receiving a doctorate in medicine in 1525. He was professor of mathematics at Milan, Pavia and Bologna leaving each after some scandal. Cardano lectured and wrote on mathematics, medicine, astronomy, astrology, alchemy, and physics. At the age of thirty four he lectured on mathematics, and at thirty five on medicine. His fame as a doctor was renown. In fact, he was so famous that the Archbishop of St Andrews in Scotland, on suffering as he thought from consumption, sent for Cardan. Cardano is reported to have visited Scotland to treat the Archbishop and cured him.

Cardano is famed for his work Ars Magna (Great Art) which was the first Latin treatise devoted solely to algebra and is one of the important early steps in the rapid development in mathematics which began around this time (and still continues today). Ars magna made known the solution of the cubic by radicals and the solution of the quartic by radicals. These were proved by Tartaglia and Ferrari respectively. Ferrari was in fact a pupil of Cardan's. We find in Ars Magna the first computation with complex numbers although Cardano did not properly understand it. The work was written completely in the rhetorical style, symbolism having not yet been invented.

Cardano's Liber de ludo aleae (1563) was the first study of the theory of probability. If anything, it is remarkable for its errors as well as truths. With Tartaglia and a century before Descartes, he considered the solution of geometric problems using algebra

Niccolo Fontana Tartaglia.

Nicolo Fontana Tartaglia (1500 - 1557) was famed for his algebraic solution of cubic equations which was published in Cardan's Ars Magna. During the French sack of Brescia (1512), his jaws and palate were cleft by a saber. The resulting speech difficulty earned him the nickname Tartaglia ("Stammerer"), which he adopted.

Tartaglia was self taught in mathematics but, having an extraordinary ability, was able to earn his living teaching at Verona and Venice (1534).

The first person known to have solved cubic equations algebraically was Scipio del Ferro. On his deathbed dal Ferro passed on the secret to his (rather poor) student Antonio Maria Fiore. A competition to solve cubic equation was arranged between Fior and Tartaglia. Tartaglia, by winning the competition in 1535, became famed as the discoverer of a formula to solve cubic equations. Because negative numbers were not used (and not even recognized) there was more than one type of cubic equation and Tartaglia could solve all types; Fior could solve only one type. Tartaglia confided his solution to Cardan on the condition that he would keep it secret, and with the implied promise of Cardano in the hope of becoming artillery adviser to the Spanish army. The method was, however, published by Cardan in Ars Magna in 1545.

Tartaglia wrote Nova Scientia (1537) (A New Science) on the application of mathematics to artillery fire. He described new ballistic methods and instruments, including the first firing tables. As well, it is a pioneering effort at solving problems of falling bodies.

Tartaglia also wrote a popular arithmetic text Trattato di numeriet misure, in three volumes (1556-60) (Treatise on Numbers and Measures), an encyclopaedic treatment of elementary mathematics. He was also the first Italian translator and publisher of Euclid's Elements in 1543. He also published Latin editions of Archimedes's works.

Ludovic_Ferrari.

From a poor family, Ludovico Ferrari (1522 - 1565) was taken into the service of the noted Italian mathematician Gerolamo Cardano as an errand boy at the age of 15. By attending Cardano's lectures, he learned Latin, Greek, and mathematics. In 1540 he succeeded Cardano as public mathematics lecturer in Milan, at which time he found the solution of the quartic equation, later published in Cardano's Ars magna (1545; Great Art). The publication of Ars magna brought Ferrari into a celebrated controversy with the noted Italian mathematician Niccolo Tartaglia over the ` solution of the cubic equation. After six printed challenges and counter challenges, Ferrari and Tartaglia met in Milan on Aug. 10, 1548, for a public mathematical contest, of which Ferrari was declared the winner. This success brought him immediate fame, and he was deluged with offers for various positions. He accepted that from Cardinal Ercole Gonzaga, regent of Mantua, to become supervisor of tax assessments, an appointment that soon made him wealthy. Later, ill health and a quarrel with the cardinal forced him to give up his lucrative position. He then accepted a professorship in mathematics at the University of Bologna, where he died shortly thereafter

Ferrari, Ludovico was orphaned at the age of fourteen. Having no formal education, he was sent as a refugee to Milan where he joined the household of Girolamo Cardano in 1536. At first he was an errand boy. Ferrari very likely showed exceptional promise, even before joining Cardano, and it is probable that this promise is what brought him to Cardano's attention. Indeed, through his lectures Cardano introduced him to Latin, Greek, and Mathematics - not the normal course of training for an errand boy. He was promoted to the post of Cardano's amanuensis, became his disciple, and ultimately collaborator. In 1540, he was appointed by Ferrante Gonzaga, the governor or Milan, public lecturer in mathematics in Milan. In so doing, he succeeded Cardano as public mathematics lecturer in Milan. In this capacity he gave lessons on the Geography of Ptolemy

He collaborated with Cardano in researches on the cubic and quartic equations, the results of which were published in the Ars magna (1545). Indeed, by all accounts, it was Ferrari who found the method of solving the quartic equation. The publication of Ars magna brought Ferrari into a well documented controversy with Tartaglia over the solution of the cubic equation. After six printed challenges and counter challenges, Ferrari and Tartaglia met in Milan on Aug. 10, 1548 for a public mathematical contest. Such challenges were common at that time as learned men sought to gain new positions or defend their existing one. The procedure was for each contestant to offer a set of problems to the other for solution. The winner was declared to be he who answered the most questions. Ferrari was declared the winner of this one.

This success brought him immediate fame and many offers for various positions. Ferrari accepted a position in the service of Ercole Gonzaga, Cardinal of Mantua, for some eight years (c. 1548 - 1556). Years later, in 1564, he returned to Bologna where he earned a doctorate in philosophy. From 1564 until his death in 1565, he was lecturer in mathematics at the University of Bologna. As an indication of his prominence, he received an offer from Emperor Charles V who wanted a tutor for his son.

Scipione del Ferro.

Scipione dal Ferro (1465 - 1525) lectured at Bologna where he was a colleague of Pacioli. Dal Ferro is the first to solve the cubic equation by radicals. He only solved one of the two cases (the fact that 0 and negative numbers were not in use made many distinct cases). He kept his discovery secret and only told his student Fior shortly before his death. Ferrari reports seeing a notebook in del Ferro's handwriting where the solution is clearly written down.