X:The Elegant Hindu Arabic Numerals

X. The Elegant Hindu Arabic Numerals
        [Contd.A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri]

So far, we have travelled a lot at different parts of the world in search of numerals and counting systems of the civilizations of the ancient times. Now let us explore the most elegant and   perfect numerals and counting system,  prevalent at preseent in whole of the world, namely, the “Hindu Arabic numerals” or in a true sense the “Hindu numerals”.

The Hindu Arabic numeral system is a unique decimal place-value numeral system which uses a specific symbol or glyph for zero. Its glyphs are descended from Brahmi* numerals. The full system emerged by the 8^th to 9^th centuries and was first described in Al-Khwarizmi’s  book "The calculation with Hindu Numerals” (825 AD) and Al-Kindi’s  four volume work “on the use of the Hindu Numerals” (850 AD). Today the name “Hindu Arabic numerals” is usually used. But these numerals were invented by the Indian Mathematicians. So the Persians and Arabic Counter parts called “Hindu numerals” where “Hindu” meant Indian. Later they came to be called “Arabic numerals” in Europe, because they were introduced to the West by the Arab merchants.

[* “Brahmi” is the modern name given to one of the oldest writing system used in South and Central Asia during the final centuries BC and the early centuries CE. Like its contemporary Kharosthi was used  which is now Afganistan and Pakistan. The best-known Brahmi inscriptions are the rock-cut edicts of Ashoka in north-central India, dated 250 – 232 BC.

Historians trace modern numerals in most languages to Brahmi numerals, which were      used around the middle of the 3^rd century BC. The place system, however, developed later. The Brahmi numerals have been found in inscriptions in caves and coins in regions near Pune, Mumbai and Uttar Pradesh in India. These numerals were in use with slight variations, over quite a long time span up to the 4^th century]

It will be worthy to know, at least briefly, who were these two scholars and renowned mathematicians of the Arab World.

Al-Khwarizmi or Latinized as Algoritmi was a Persian mathematician, astronomer and geographer, a scholar in the House of Wisdom in Baghdad. In the 12^th century, Latin translations of his book on Indian numerals introduced the decimal positional number system to the Western World.

And Al-Kindi known as “the philosopher of the Arab’s” was a Muslim Arab philosopher, polymath, mathematician, physician and musician. He was born in Basra and educated in Baghdad, two well-known Iraqi cities. Al-Kindi became a prominent figure in the House of Wisdom and he was entrusted with the job of translations of Greek scientific and philosophical texts into Arabic language. It enriched him greatly with Greek philosophy. In the field of mathematics Al-Kindi played an important role in introducing Indian numerals to the Islamic and Christian World.

During Vedic period (1500 – 500 BC), motivated by geometric construction of the fire altars and astronomy, the use of a numerical system and basic mathematical operations developed in northern India. Hindu Cosmology required the mastery of very large numbers, such as ‘Kalpa’, the life time of the universe and said to be 4,320,000,000 years. The expanse or ‘orbit of the universe’, as they calculated to be 18,712,069,200,000,000 yojanas, according to modern measure a yojana is a distance about 8 km, controversially may be a distance between 8 and 13 km. So, what an unimaginably large number they could think of!

Numbers were expressed using names for the place-value in powers of 10 as dasa, satha, sahasra, ayuta, nijuta, prajuta, arbuda, nyarbuda, samudra, madhya, anta, parardha etc. These places, as per our modern notation in powers of 10 were 10¹, 10², ------ upto 10¹², the last of these being the name for trillion. For example, the number 46352 was expressed as 4 ayuta six sahasra 3 satha 5 dasa 2. The extreme right hand numeral 2, as per modern usage, is in unit’s place having a place value of 10⁰ or 1 and is a multiple of 1.

It is a long history how the present form of glyphs or symbols had come into prevalence. Various symbol sets were used to represent numbers in Hindu-Arabic numeral system, all of which were developed from Brahmi numerals. The symbols used to represent the system had split into various typographical variants since the Middle Ages may be classified into three main groups as shown in Table No.-9a.

Table No.9a                                                      

                                                     Brahmi numerals.

Firstly, the wide spread Western “Arabic numerals” used with Latin, Cyrillic and Greek alphabets, sometimes called “European” descended from the “West Arabic numerals”.

Secondly, the “Arabic-India” or “Eastern Arabic numerals” used with Arabic scripts was developed primarily in what is now Iraq.

Thirdly, the “Hindu numerals” in use with scripts of the Brahmi family was in India and South East Asia. Each of the roughly dozen major scripts of India has its own numeral glyphs.

Some of those major glyphs are as the followings: [Table No. : 9(b) and : 9(c)]

Table No. : 9(b) : Numeral glyphs descend from Brahmi family

Table No. : 9(c) : Hindu Arabic numerals

It is without doubt that mathematics today owes a huge debt to the outstanding contributions on beautiful number system invented by the Indian mathematicians over many hundred years, on which much of mathematical development rested. What is quite surprising is that many famous historians of mathematics are reluctant to recognize this, though it was so clear in front of them.

The history of earliest development of Indian mathematics may be traced back to Indus valley civilization flourished about 2500 BC and survived until 1700 BC or later. This earliest known urban Indian culture was first identified in 1921 at Harappa and then one year later at Mohenjo-Daro, near the Indus River in Sindh. Both of these are now in Pakistan but this is still covered by the term “Indian mathematics”.

The next mathematics of importance on the Indian subcontinent was associated with the religious texts of Vedas which were composed in Vedic Sanskrit between 1500 BC and 800 BC. At first these texts, consisting of hymn, spells and ritual observations were transmitted orally. Later the texts became written works for the use of those practicing the Vedic religious. It consisted of the Sulbasutras which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed not for its own sake, but purely for religious purposes. The mathematics contained in these texts is studied in some detail in the separate article on Sulbasutras.

Ancient and medieval mathematical works, all composed in Sanskrit, usually consisted of a section of Sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary that explained the problem in more detail and provided justification for the solutions. All mathematical works were transmitted orally until approximately 300 BC; there after they were transmitted both-orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is “Bakshali Manuscript”*

[*The Bakshali Manuscript is a mathematical script found by a peasant in the village of Bakshali which is near Peshawar, now in Pakistan, in 1881. It is notable for being the oldest extant, likely from the 7^th century CE, manuscript of Indian mathematics. The manuscript is incomplete consisting of seventy leaves of birch bark. The intended order of those leaves is indeterminate. It is currently housed in the Bodleian Library of the University of Oxford.

The manuscript is a collection of rules and illustrative examples. The sample problems are in verse and the commentary is in prose associated with calculations and solutions. The problems involved arithmetic, algebra, geometry including mensuration. The topics covered include fractions, square roots, arithmetic and geometric progressions, solution of simple equations, simultaneous linear equations, quadratic equations and indeterminate equations of second degree.]

The development of positional decimal system takes its origin in Hindu Mathematics during Gupta period*. Around 500 AD, the astronomer Aryabhata used the word “Kha”, meaning emptiness, to mark zero in tabular arrangements of digits. The 7^th century Brahamasphuta Siddhanta contained a comparatively advanced understanding of the mathematical role of zero. The first dated and undisputed inscription showing the use of a symbol for zero appears on a stone inscription found on the Chaturbhuja Temple at Gwalior in India, dated 876.

[* Gupta Empire : Gupta Empire was an ancient Indian empire founded by Sri Gupta which existed at its zenith from approximately 320 to 550 CE and covered much of the Indian subcontinent. The peace and prosperity created under the leadership of the Guptas enabled the persuit of scientific and artistic endeavours. This period is called the Golden Age of India and was marked by extensive inventions and discoveries in science, technology, engineering, art, dialectic, literature, logic, mathematics, astronomy, religion and philosophy that crystallized the elements of what is generally known as Hindu Culture.

Chandragupta – I, Samudragupta and Chandragupta – II were the most notable rulers of the Gupta dynasty. The Guptas conquered about 21 kingdoms, both in and outside India. The 4^th century CE legendary Sanskrit poet Kalidasa was one of the nine scholars, known as “Navaratnas” of the court of the legendary ruler Chandragupta – II, famously known as Vikramaditya.]

Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, algebra. In addition, trigonometry was further advanced in India.

In a more developed form of the Hindu numeral positional decimal system, there was also use of decimal marker, at first a mark over one’s digit, but now more usually a decimal point. Moreover a symbol for what we now call “recurring decimal” or to specify “these digits recur ad infinitum” was used. In modern usage, this later symbol is usually a vinculum, a horizontal line placed over the recurring digits. In this more developed form, the numeral system can symbolize any rational number, namely, a number which can be expressed as a ratio of two numbers, using only 13 symbols. Those are ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), decimal marker, vinculum and a prepended dash to indicate a negative number.

Indian mathematics emerged in Indian subcontinent from 1200 BC until the end of the 18^th century. In classical period of Indian mathematics ranging from 400 CE to 1600 CE important contributions were made by scholars* like Aryabhata, Varahamihira, Brahamagupta, Bhaskara-I, Bhaskara-II. Madhaba of Sangagrama, Nilkantha Somayaji and many others.

[*Aryabhata : The first most important name of mathematician and astronomer of early classical period of Indian mathematics is Aryabhata (476 – 550 CE).He is also known as Aryabhata – I or Aryabhata the Elder to distinguish him from a 10^th century Indian mathematician of the same name. The theories that he came up at that time present a wonder to the scientific world today. His works were used by the Greeks and the Arabs to develop further.

Aryabhata born in 476 CE but provides no information about his place of birth. According to scholars he was born in Tarenaga, a town in Bihar, India. It is however definite that he travelled to Kusumapara which is modern day Patna for studies and flourished there. It is mentioned in a few places that Aryabhata was the head of the educational institute in Kusumapara. The famous University of Nalanda had an observatory in its premises and it is hypothesized that Aryabhata was the principal of the university as well. On the other hand some other commentaries mention that he belonged to Kerala. But this hypothesis has no strong evidence.

Aryabhata wrote many mathematical and astronomical treatises. His chief work was the ‘Aryabhatiya’ which was a compilation of mathematics and Astronomy. It covers several branches of mathematics such as algebra, arithmetic, plane and spherical trigonometry. Also included in it are theories on continued fractions, sum of power series, sine tables and quadratic equations.

Aryabhata worked on place value system using letters to signify numbers and stating qualities. He also came up with an approximation of pi (π) which is the ratio of the circumference to its diameter of a circle. He introduced the concept of sine in his work called ‘Ardha-jya’ which is translated as ‘half-chord’.

Aryabhata also did a considerable amount of work in astronomy. The Aryh-Siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhat’s contemporary, Varahamihira and later mathematicians and commentators, including Brahamagupta and Bhaskara – I. He knew that the earth is rotating on its axis around the Sun and the moon rotated around it. This heliocentric idea was not known until the 16^th century, more than a millennium later, when a geometric mathematical model of a heliocentric system was presented by Renaissance mathematicians and astronomers and a Polish Catholic Cleric Nicholus Copernicus, leading to the Copernican Revolution.

In some texts, he seems to ascribe the apparent motions of the heavenly bodies to the Earth’s rotation. He also discovered the position of the nine planets and stated that these also revolve around the Sun. He may have believed that the planets’ orbits as elliptical rather than circular which was discovered much later, in 1609, by a famous German mathematician and astronomer Johannes Kepler.

He scientifically explained the eclipses, both solar and lunar. He stated that the moon and planets shine by reflected sunlight. Instead of prevailing Cosmology based on common belief that eclipses are caused by Rahu and Ketu, two demons, he explained the eclipses in terms of shadows cast by and falling on Earth.

Considered in modern English units of time, Aryabhata calculated the diurnal rotation or time of full rotation of Earth about its axis with respect to the Sun (Sidereal rotation) as 23 hours, 56 minutes, 4.1 seconds, while the modern value is 23:56:4.091.

Similarly, his value for the time Earth takes to complete its orbit around the Sun with respect to fixed stars (Sidereal year) was 365 days, 6 hours, 12 minutes and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds (365.25636 days); astonishing! Unbelievable! What a genius Aryabhata was! India’s first satellite launched in 1975 was named after this great astronomer Aryabhata.

He was the first person to mention that the Earth was not flat but in a spherical shape. He also gave the circumference and diameter of the Earth and radius of orbits of planets.

Brahama Gupta :

Brahamagupta (598 – 670 CE) was one of the most significant mathematicians of ancient India. He introduced extremely influential concepts to basic mathematics, including the use of zero in mathematical calculations and the use of mathematics and algebra in describing and predicting astronomical events.

Brahamagupta was born in 598 AD in Bhinmal, a state of Rajasthan in India. He spent most of his life in Bhinmal. He was the head of the astronomical observatory at Ujjain which was the centre of mathematics in India witnessing the work of many extra-ordinary mathematicians. The astronomically significant ancient Indian city of Ujjain was a place near the tropic of cancer that occupies a place in Indian history somewhat comparable to that of Greenwich in England. It was a central reckoning point for ideas of time and space, and it became a major astronomical and mathematical centre.

Little else is known of the life of this mathematician and astronomer who flourished 1400 years ago. It is known that, though he was a devout Hindu, yet he however, rejected the ancient Hindu ideas that the Earth was flat or bowl-shaped; like ancient Greek thinkers, including Aristotle, he realized that it was a sphere and rotate about an axis.

Brahamagupta is known mostly through his writings, which cover mathematical and astronomical topics and significantly combine the two. The first of his two surviving treaties was “Brahama Sphuta – Siddhanta”, often translated as “The opening of the universe” was written in 1628 when he was about 30 years old. His second, the “Khandakhadyaka” which means something like “Edible Bite”, is less well known. It expands on the work of the earlier astronomer, Aryabhata, whose chief contribution was the idea of beginning of each day at midnight. It was written in 665 AD, near the end of his life.

Brahamagupta’s first manuscript the Brahamasphutasiddhanta, was a revision of an older astronomy book “Brahamasiddhanta”. It opened with three chapters on the position and motions of the planets and stars, and on the cycle of day and night. Two chapters dealt with lunar and solar eclipses respectively and one with heliacal rising and setting of stars, planets and moon – the seasonal reappearances and disappearances of these celestial bodies as they pass the horizon line before being hidden by the sun. He also discussed phases of the moon, close approaches or conjunctions of planets in the sky and conjunctions between planets and stars. Brahamagupta also calculated the length of the solar year as 365 days, 6 hours, 5 minutes, and 19 seconds, among the most accurate of early reckonings and remarkably close to actual value of 365 days, 5 hours, 48 minutes and about 45 seconds. It should be remembered that these were remarkable estimates in an era that had no telescope or scientific instruments in the modern sense.

After discussion of astronomy, Brahamagupta turned to mathematics, discussing what would now be called arithmetic and algebra which he termed as “pati-ganita”, or mathematics of procedures, and “bija-ganita”, or mathematics of equations. These ideas laid the foundation for much of the later development of mathematics in India. Some of Brahamagupta’s discussions will sound familiar to the modern students of mathematics. His direction for multiplication of large numbers, in a sense, is close to what students are taught today. His preferred multiplication method, according to the mathematics history website maintained by St. Andrews University in Scotland, is given the name “gomutrika” by Brahamagupta, meaning “like the trajectory of a Cow’s Urine”.

 Brahamagupta also introduced new methods for solving quadratic equations that will be recognizable to modern students of mathematics. He also devised formulas for calculating the area and length of the diagonals of a cyclic quadrilateral. His method is still known as Brahamagupta’s theorem. Brahamagupta investigated various higher functions of algebra and geometry, in each case building on and refixing the mathematical heritage of the ancient world. A curious feature of Brahamagupta’s treatise is that it is largely written in Sanskrit verse.

Although it is difficult to pinpoint single inventor of the concept of zero, Brahamagupta is a reasonable contender for the title. Perhaps Brahamagupta’s most important innovations, however, pertained to his treatment of the number zero. Several different discoveries converged to from the concept of zero. The circular symbol for the number and the idea of representing orders of magnitude in a number through the use of places arose at different times and places prior to Brahamagupta’s work. Brahamagupta, however, was the first to propose rules for the behavior of zero in common arithmetical equations, relating to zero to positive and negative number which he termed ‘fortunes’ and ‘debts’. He correctly stated that multiplying any number by zero yields a result of zeros but erred, as did many other ancient mathematicians, in attempting to division by zero. Nevertheless, Brahamagupta is sometimes called the “Father of zero”.

The impact of Brahamagupta’s discoveries was felt in Islamic world. Brahamagupta’s writings were translated into Arabic in 771 and they had a major impact on subsequent writers in the Arab world, including al-Khwarizmi, the ‘father of algebra’. The mathematical thought of medieval and early modern Europe was influenced by Arabic models that had been in existence for centuries. Distant from the modern mathematics in time and place, Brahamagupta nevertheless exerted a definite influence on mathematics as the discipline is known today.

Bhaskaracharya : (1114 – 1185 CE)

Bhaskaracharya is also known as Bhaskara – II to avoid confusion with the same name of mathematician of the 7^th century AD. He was a great Indian mathematician and astronomer. Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12^th century. He has been called the greatest mathematician of medieval India.

He was born in 1114 CE near Vijjadavida, believed to be Bijjaragi of Bijapur in modern Karnataka. Bhaskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical centre of medieval India. It goes to the credit of Varahamihira and Brahamagupta, the leading mathematician who worked there and built up this school of mathematical astronomy.

Bhaskaracharya’s main work Siddhanta Shiromani consists of four parts called Lilavati, Bijganita, Grahagamita and Goladhyaya, which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of planets and spheres respectively. He also wrote another treatise named Karana Kautuhala.

It is surprising that Bhaskara-II had conceived the basic ideas of differential calculus, a brilliant and effective mathematical tool for future development of applied sciences in general and mathematics in particular, predates Newton in England and Leibnitz in Germany half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibnitz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus.

He wrote Siddhanta Shiromani at the age of 36. This colossal work consists of about 1450 verses. Each part of the book consists of huge number of verses and can be considered as a separate book. Lilavati has 298, Bijganita has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses. He formulated the simple ways of calculation from Arithmetic to Astronomy in this book. He wrote Lilavati in an excellent lucid and poetic language. It has been translated in various languages throughout the world.

                                                                                                                      [To continue]

  Reference: Internet.

 Image credit:Table No 9a: Bramhi numerals: Creative Cosmos (https://commons.wikimedia.org)/wiki/File:Brahmi_numeral_signs.svg)
                                      9b:Glyphs of numerals descended from Bramhi:                                           Wikipedia (https://en.wikipedia.org/wiki/History_of_science_and_technology_in_the_Indian_subcontinent )
                                       9c:Hindu Arabic numerals: ( http://archimedes-lab.org/ )
                          

IX:The ancient numerals of the Mighty Romans

IX.  The ancient numerals of the mighty Romans

   [Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri]

          

           We have heard many fascinating stories about ancient Rome and of mighty Roman emperors.Their ancient numerals also have importance even today and continues to be widely used,however,in certain formal  and minor contexts.For examples:on clock faces showing hours,coins,for inscriptions on corner foundation stones,generation suffixes such as Louis XIV,King George VI etc.

Let us have a glance at the ancient Roman civilisation before going to search for their numerals. The great ancient Roman civilisation is undoubtedly one of the mightiest and most influential of all cultures that came before and after.

Rome sprung up as a small, agricultural civilisation in the 10^th century BC from a small town in central Italy’s Tiber river in the 8^th century BC supposedly founded by the mythological twins Remus and Romulus. The traditional date of foundation of Rome is 21 April 753 BC. The origin of the city’s name “Rome” is thought to be after the name that of the reputed founder and the first ruler, the legendary Romulus. As the legend goes, Romulus and his twin brother Remus, apparent illegitimate sons of the Roman war god Mars and the descendants of the Trojan hero Aneas, were suckled by a she-wolf after being abandoned. After growing up they decided to build a city. The brother’s argued, Romulus killed Remus and then named the city Rome after himself.

Beginning in the 8^th century BC, ancient Rome grew up from a small town on central Italy’s Tiber River into an empire that at its peak encompassed most of continental Europe, Britain, much of Western Asia, Northern Africa and Mediterranean islands.  After 450 years of republic, Rome became an empire in the wake of Julius Caesar’s rise and fall in the first Century BC. The long and triumphant reign of the first emperor Augustus, began a golden age of peace and prosperity.

As the Roman empire grew, it became more and more difficult to rule from the city of Rome. To solve this problem, the Roman leaders decided to split it to two empires. Accordingly, one was the Western Roman Empire, to be ruled from the city of Rome itself, the other part was Eastern Roman Empire or Byzantine Empire which was to be controlled from the city of Constantinople, founded by Constantine the Great in 330 AD. At present we know this city as Istanbul, an important city of Turkey.

The fall of Rome generally refers to the fall of the Western Roman Empire. It fell in 476 AD. However, the Eastern Roman Empire or the Byzantine Empire ruled parts of the Eastern Europe for another 1000 years. The empire’s decline and fall by the fifth century AD was one of the most dramatic implosions in the history of mankind.

Ancient Rome was the progenitor of Western civilization. Among many legacies of Rome dominance are the wide spread use of Roman Language that derived from Latin such as Italian, French, Spanish, Portuguese and Romanian; the modern western alphabet, calendar and emergence of Christianity as a major world religion. Many more aspects of Roman civilisation worthy to mention are : Customs, law, technology, architecture, political system, military, literature, government and many factors and aspects of Western Civilisation are all inherited from Roman advancement.

In ancient Roman Society, women were accorded an important place. They enjoyed and shared almost equal right with Roman men and were provided similar opportunities to excel in education, business and trade.

‘Slavery’ is a contemptuous word to all of us now-a-days, as it grossly violets human rights, but in ancient Rome, slavery played an important role in society and economy. Romans, especially the rich ones depended greatly on their slaves for maintaining a luxurious lifestyle. Besides manual labour, slaves performed many domestic services and also be employed at highly skilled jobs and professions like the job of an accountant and physicians. Greek slaves in particular might be highly educated. Unskilled slaves or those sentenced to slavery as punishment, worked on farms, in mines and mills. Their living condition was brutal and their lives were short. They were considered properties under Roman law.

Roman Empire owed much of its achievements and grandeur to highly professional and disciplined Roman military. They achieved wonderfully in science and architecture at those days.

One of the greatest scientific feats of ancient Rome is the concrete roads. Nearly 30 military highways, all made of stone, exited in the great city. At one point, 372 roads connected 113 provinces. It helped greatly in efficient military operations and also in improvement in social life, thereby furthering civilisation.

Like art and other aspects of ancient Roman culture, early Roman architecture borrowed heavily from Greek architecture. However, they quickly adopted their own style and marveled in this field.

If you are to visit Rome, you could see many of the ancient buildings even today such as the Forum, the Colosseum and also many architectural marvels in domes like the Roman Pantheon, Circus Maximus and also famous aqueducts and bath like the Baths of Diocletian etc.

The Forum was at the centre of the city and the Roman public life was centred about it. This was a rectangular plaza surrounded by public buildings such as temples of gods and basilicas where commerce and other public functions could take place.Here many of the city’s major events, such as elections, trials, public speeches and triumphant processions were held. At the centre of Rome there were many other famous buildings like temple of Jupiter, the Colosseum, Circus Maximus, the Roman Pantheon and Pompe’s theatre.

The Colosseum is the largest amphitheater ever built and is one of the Seven wonders of the world. This is situated just east of the Roman Forum. As it is estimated, the Colosseum could hold between 30,000 and 60,000 spectators having an average audience of about 65,000. It was used for gladiatorial contest and public spectacles such as mock sea battles, animal hunt, executions, reenactment of famous battles and dramas based on classical mythology. Of all the functions held in the Colosseum, the gladiatorial contest was the worst adventurous, barbaric and most fearful. A gladiator was an armed combatant who entertained audience in the Roman Republic and Roman Emperors in violent, confrontations with other gladiators, wild animals like lion, tiger etc. and condemned criminals. Most were despised as slaves, schooled under harsh conditions, socially marginalised and segregated even in death.

Another architectural excellence of ancient Rome is the Roman Pantheon, the largest dome in the world for more than a millennium. It is the largest unreinforced solid concrete dome to this day.

The Circus Maximus was also a wonder of ancient Rome. It was a huge chariot racing stadium which could hold around 150,000 spectators. It was built on the land between Palatine and Aventine hills that two hills of the seven hills on which Rome was built.

Roman aqueducts and baths are example of wonderful advancement of ancient Roman technologies. The Romans constructed numerous aqueducts in order to bring water from distant sources into cities and towns, supplying public baths; latrines, fountains and private households. Waste was removed by complex sewage systems and released into nearby water-bodies, keeping town clean and free from effluent. Aqueducts also provided water for mining operations, milling, farms and gardens.

Romans also built many baths for public uses. Baths of Diocletian, named after the emperor Diocletian, was the largest and most impressive imperial baths. Of course you now could not find them in Rome. Parts of the structures were used for different purposes. The basilica of Santa Maria stands on the remains of the baths.

Roman art was an important aspect of ancient Roman civilisation. Roman art refers to the visual arts in ancient Rome and in territories of the Roman Empire. It includes architecture, painting, sculpture and mosaic work. At present luxury objects in metal work, gem engraving, ivory carvings and glass are sometimes considered to be minor forms of ancient Roman art. Sculpture was perhaps considered as the highest form of art by Romans, but figure painting was also very highly regarded. It is thought that the ancient Roman artists often borrowed from and copied Greek precedents; for much of the Greek sculpture known to-day is in the form of Roman marble copies. But also the Roman artists had their own contributions in art. A large body of ancient Roman Sculptures surviving from about first century BConwards, though little from before, but very little painting at all remains which may not be considered to be of the highest quality.

To have a glimpse at the ancient Roman paintings, let us go to the Villa of the Mysteries, a well preserved suburban Roman Villa on the outskirts of Pompei, Southern Italy. It is famous for the frescos in one room which are now probably the best known of the relatively rare survivals of ancient Roman painting. Like the rest of the Roman city of Pompei, the Villa was buried in the eruption of Mount Vesuvius in 79 AD and excavated from 1909 onwards. It is how a popular site of tourist visits to Pompei and forms part of UNESCO World Heritage Site at Pompei.

It has already been mentioned that the ancient Roman civilisation was one of the mightiest and most influential of all ancient cultures. In fact, ancient Rome was the progenitor of western civilisation but they had no substantial contribution towards mathematics. They invaded many kingdoms and countries including Greece. The Greeks at that time was much advanced in science, mathematics, art and philosophy than the Romans. Roman invasion of Greece was only for power, not for abstract mathematics. They killed Archimedes, the renowned Greek scientist, in 212 BC and thereby impeded the development of mathematics. On the other hand, the system of Roman numerals was too complicated for calculations. Actual counting had to be done on a counting board , an early form of abacus although the usage of Roman numeral system spread all over Europe and remained dominant  numeral system for more than five hundred years, yet not a single Roman mathematician is celebrated to-day. The Roman were more interested in using numbers to record their conquest and to count dead bodies.

Roman numerals follow the history of ancient Rome itself from its early stages at Latin Palatine Hill in the 8^th and 9^th century BC to its fall in the 2nd century AD from civil war, plague, civil apathy, rise of Christianity and Northern European powers.

There exist many theories as to the origin of the numerals; however it is widely accepted that Roman numerals are based upon a more ancient Etruscan system. Etruscans, who lived in central Italy before the Romans, developed their own numeral system with different symbols.

One theory of ancient Roman numerals is that of the Tally Sticks made of wood or bone. Tally sticks or some similar methods of simple recording had been in use long before founding of the Roman Empire. During ancient times merchants used tally sticks for recording business transactions. On tally sticks, single digits were represented with a single notch, where as a group of five was represented as either a standard or inverted V. To represent the value of 10, a tally stick simply crossed two single notches X.

An alternative to the tally stick theory is that Roman numerals are a representation of hand signals. The Roman numerals I, II, III, IIII represent the number of fingers held up during auctions when merchants would gather together and bid against each other, similar to modern day auction. This hand signal theory suggests that the Roman numerals is a direct representation of these hand signals. For larger numbers such as 5 or 10 merchants would hold up their hand with their thumb extended and their fingers together to represent 5 or arms crossed for 10, which are similar to V and X used in Roman numeral system.

Roman numerals are represented by seven different letters: I, V, X, L, C, D and M which represent 1, 5, 10, 50, 100, 500 and 1000 respectively. In fact, the Romans built their numerals based upon these Latin alphabets utilizing a series or combination of letters which represent numeric values. Roman numerals are an additive and subtractive system or more clearly, a system that add or subtract symbols in order to make corresponding numbers.

Roman numerals were usually written from largest to smallest , from left to right, this tells us that we must add the numerals together. However, this is not always the case. Romans didn’t like four of the same numerals to be written in row, so they developed a method of subtraction.

So, in Roman numeral system the number 2 will be represented by II which means two ones are combined together. Similarly 3 will be written as III, 12 as XII, that means 12=10+2 or X+II, 28 will be XXVIII, implies 28=20+5+3 as XX+V+III etc.

Likewise, by the subtractive system the number 4 will be 5 – 1 or IV, 29 will be written as 30 –1=IXXX, 90 as 100 – 10=XC etc. There are six instances where the Romans used subtraction, namely I can be placed before V(5) and X(10) to make the numbers 4 and 9.

X is placed before L(50) and C(100) to make 40 and 90.

C to be placed before D(500) and M(1000) to make 400 and 900.

A good example of the above rules, the representation of 1904, in Roman numerals

For 1904 = 1000+900+4 will be written as M+CM+IV or MCMIV

But it is difficult to write large numbers only with these Seven letters, the largest being M (1000). In fact, Roman rarely needed to write numbers larger than 3999 and that is why they never developed a system to overcome this problem, although a system was developed much later to do so.

The Roman number system flawed in many ways. The greatest of them was they had no idea of Zero (0) as a place holder yet, the word “nulla” meaning “none” in Latin was used by medieval computists in lieu of zero and sometimes represented by N, the initial letter of Nulla.

Though Romans used a decimal system for whole number, yet they used a duodecimal system for fractions. The reason behind it was that 12 has many divisors, like 2,3,4, and 6 while 10 has only two, 2 and 5. Divisibility of 12 makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on 10.

Romans used coins which were fractions from 1/12 to 12/12, the unit called “as”. They used a tally – like notational system based on twelfths and halves. One twelfths was called “uncia” which was indicated by a dot (.). These dots were repeated for fractions up to five twelfths. Six twelfths or one-half was abbreviated by S, the first letter of the word ”Semis”, meaning half. Uncia dots were added to S for fractions from Seven twelfth to eleven twelfths, just as tallies were added to V for whole numbers from six to eight.

Each fraction from 1/12 to 12/12 had a name in Roman times which corresponds to the related coins. Are you curious to know those names? It is given below. You will be surprised to discover the source of the English words “inch” and “ounce” from this table.

Table No. 8

Here one thing is to be noted that the arrangements of dots was variable and not necessarily linear. Five dots arranged like (∵) as on the face of a die are known as quincunx, derived from the name of the Roman fraction or Coin. The Latin words sextans and quadrans are the source of the English words Sextants and Quadrants. The list of Roman fractions above is not exhaustive.  There were other Roman fractional notations also.

Another usage of Minuscule or lower case letters of the Roman numerals was developed, well after fall of the Western Roman Empire, in Middle Ages and renaissance. Since that time lower case versions of Roman numbers have also been commonly used.  These numerals are as i, ii, iii, iv and so on.

Since the Middle Ages, a “j” has sometimes been substituted for the final “i” of a lower case Roman numeral, such as iij for 3 or vij for 7. Perhaps to signify the final unit to stop tampering and also to avoid misinterpretation of a number after it had been written on a document, j is used for the final i. We still find this practice in writing medical prescriptions.

Absence of Zero in Roman numerals hindered their advance in mathematics. The Romans probably didn’t use numerals for complete arithmetical operations due to its complexity. In this system the operation of multiplication and division are far too impractical. However, it is not impossible for addition and subtraction.

But before going to perform the operation of addition with Roman numerals, it is important to note that we don’t use the subtractive system of writing the numbers to be added. For example, the number four is not shown as IV but as IIII.

Let us take a simple case of addition of IX and XIII. Here we must first change IX to VIIII, as it uses subtractive principle. So, now we are left with  VIIII  + XIII. Now we can complete the sum just like we would if we were using Arabic numbers as shown below.

V I I I I         +  X I I I ――————— X V I I I I I I I

We arrange the numerals from biggest to smallest which gives XVIIIIIII. The next

step is to simplify IIIIIII to V which gives us VII and the sum as XVVII. This can further be simplified as XXII or 22, the required sum.

Similarly, in the process of subtraction we can’t use subtractive principle of writing the numerals. For example, let us subtract CCLXXI from CCLXXXXVIII or CCLXXXXVIII – CCLXXI. After cancelling the common characters CCLXX from both numbers, it gives   CCLXX XXVIII –  CCLXX I = XXVIII – I which can further be simplified as XXVII which the required result. In our Arabic numeral system, this problem actually is : to subtract 271 from 298. The required result is 298 – 271 = 27 or XXVII in Roman numerals.

Various uses of the ancient Roman numerals can still be seen in modern day. In fact they are all over the place!

Roman numerals are widely used to refer kings, queens, emperors and popes. For example, George V was the king of England to be called George. By using numeral we can easily distinguish between which King George we are talking about. Similarly, Queen Elizabeth-II or Pope Paul – VI refer to specific queen and pope respectively.

Many competitions such as in Olympic Games, use of the numerals to signify how many times the event have preceded the current one. For example, the Olympic Games in Brazil in 2016 will be the 31^st Games, so it will be the XXXI Games.

Roman numerals can often be found on buildings and monuments to illustrate the year of construction. Say, for instance, if we see the numerals MMIV have engraved on a building wall, we at once understand that the building was built in the year 2004.

Many movies use these Roman numerals to illustrate in which year the film was copyrighted or created. For an example : the movie ‘Gladiator’ was  copyrighted in the 2000. So it may have the numeral MM at the end of the credits or the film ‘Spartacus’ which was copyrighted in the year 1960 will have the numeral MCMLX.

A lot of clocks, even today, use Roman numerals to represent hours. We find frequently on the faces of clocks, for example, 3, 6, 9 and 12 o’clock are depicted as III, VI, IX and XII respectively.

Ancient Roman numerals can be found in so many real life situations. The list is not exhaustive, it goes on and on; at the beginning of books to number the pages before the book actually begins or within legislation to label sections, sub-sections etc. and even to refer to wars; such as WWI and WWII.
 Attribution:Table No.8----Ancient Roman Fractions
Creative Cosmos ( https://en.wikipedia.org/wiki/Roman_numerals#Fractions )
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VIII:Alphabetic numerals of the Greeks

VIII : Alphabetic numerals of the Greeks

     

         [Contd. A Journey to the Wonderland of Math. by Ajay Kumar Chaudhuri.]

Now, let us explore another famous ancient civilisation, famous for its valuable contribution of mathematical principles and for their unique numerals. This is the Greek civilisation, which flourished in the 8^th century BC and it is normally said the glorious period ended when Greece fell to the Romans in 146 BC.

Greece is considered to be the cradle of Western civilisation, having been the birth place of democracy, Western philosophy, the Olympic Games, Western literature, historiography, political science, major scientific and mathematical principles and western drama, including both comedy and tragedy. The Greeks were first unified under Philip of Macedon in the fourth century BC. His son Alexander the great rapidly conquered much of the ancient world, spreading Greek culture and science from the eastern Mediterranean to the Indus River. Annexed by Rome in the second century BC, Greece became an integral part of the Roman Empire and its successor, the Byzantine Empire.

Greece is strategically located at the crossroads of Europe, Asia and Africa. Situated on the southern tip of the Balkan Peninsula, it shares land borders with Albania to the northwest, the Republic of Macedonia and Bulgaria to the north and Turkey to the north-east.

To ancient Greeks, their alphabets had two roles to play: One for writing and the other to express numerals. The Greek numbering system was uniquely based on their alphabets. But there were no single Greek national standards in the first millennium BC, since the various island states adhered to their individual independence. Naturally they each had their own currency, weights, measures etc. and there by it lead to small differences in the number system among different States. However, we may ignore those small differences but rather we will look at its general structure. It is also remarkable that ancient Greeks used different systems for expressing cardinal and ordinal numbers. (A cardinal number is a number that says how many of something are there, such as one, two, three, four etc. Whereas an ordinal number is a number that tells the position of something in a list such as 1^st, 2^nd, 3^rd, 4^th, etc,), So we must look carefully at what we mean by Greek number system.

How Greeks used their alphabets to symbolise numbers are shown in the Table no.5 below with their sounds, using lower case ones, as it is easier to recognize, though strictly speaking is inaccurate.

Table No.5

It is evident from the table that 27 numbers to be alphabetically represented. But there are only 24 classical Greek alphabets. So, they had to find 3 extra symbols for the missing numbers of 6, 90 and 900. To supplement this, they used 3 archaic or obsolete letters which used to be in the alphabet but had been dropped as they were no longer required.

Since both the upper and lower case versions of 24 classical Greek letters were used to symbolise numbers, it is important to be familiar with both the versions of those classical letters. They are as follows:

    Table No-6

                                       Greek alphabets upper and lower cases.

This alphabetic numerals of the Greeks was hard for arithmetical calculations, for someone had to remember that alpha + alpha = beta, iota + iota = Kappa, rho + rho = sigma etc. What a lot of symbols to keep track of! Multiplication in the Greek system was even worse. They would have to use abacuses for arithmetical calculations.

As the same alphabet may represents a letter as well as a number, so there was   a chance of ambiguity for using the same letters for making words as also for numbers. The Greeks sometimes enjoyed this as a fun with letters.

  Suppose the name of a young girl is Airlea. So in Greek letters it would be αւρλεα          whose numeric value is 1+10+100+30+5+1=147. But it will be very difficult to say, what are the names whose numeric value would be 147; for 147 may be split in many many number of ways whose sums will be 147 and hence may represent a host of names.

In the book of Revelations, a book of the New Testament that occupies a central place in Christian eschatology, we find, “Let him that hath understanding count the number of the beast for it is the number is Six hundred three score and Six”. There had been many suggestions as to what the name might be that would produce this value of 666. The Roman Emperor Nero was the most likely candidate for his boundless brutality. The full name of this emperor was Nero Claudias Augustus Germanicus and the sum of the numeric values of the word “Claudias” is 20+30+1+400+4+10+1+200 which is exactly 666.

Although the Greek alphabet began with only majuscule or upper case form (using capital letters), surviving papyrus manuscripts from Egypt show that unical or majuscule and cursive minuscule (lower case form or small letter form) forms began early. These letter forms sometimes replaced the former ones, especially in the case of the obscure numerals. For example, the numeral for 6 changed several times.

In Modern Greek, a number of other changes have been made. Instead of using overbar over an entire number, the Keraia is marked to its upper right, a development of the short marks formerly used for single numbers and fractions.

Now let us have a look at the first Greek numerals system dated back to first millennium BC. This was acrophonic system which means that the symbols for numerals come from the first letter of the name of the numbers. So the symbol has come from an abbreviation of the word which is used for the numbers.

In Greek language the numbers five, ten, hundred, thousand and ten thousand are Pente, Deka, Hekaton, Khilioi and Murioi respectively. So the acrophonic forms of those numbers are as in Table No.7(a).
    Table No.7a.

                                              Acrophonic forms of some Greek numbers.

In this acrophonic system there were no symbols for the numbers from 1 to 4, obeying the above rule, however, these numbers were expressed by simple strokes like I, II, III, and IIII which was an obvious notation not coming from the initial letter of the name of a number. So, the numbers from 1 to 10 in acrophonic system based on additive system will be as in Table No.7(b).

             Table No.7b.

                                              

There awaits one puzzle for us for the symbol of 5 which should be P, if it was derived from the first letter Pente. However this is simply a consequence of changes to Greek alphabet after the numerals coming from these letters had been fixed. By that time the symbols for numbers were probably not thought of as coming from the letters so there was no attempt to change them with changes to the symbols for the letters. One interesting thing to note here that the original form of π (pi) was G and so ,Pente was Gente.

But there were some short comings of this system. If base 10 is used with an additive system without intermediate symbols then many characters are required to express certain numbers. For example the number 9999 would require 36 symbols in this system which is very cumbersome.

In Greek acrophonic numbers there was a special symbol for 5, which undoubtedly cut down characters for representing numbers and perhaps arose from counting on fingers, as we have 10 fingers and 5 on each hand. What is a bit surprising is that the system had intermediate symbols for 50, 500, 5000 and 50000 but they were not new characters, rather they were composite symbols made, from 5 and the symbols for 10,100, 1000 and 10000 respectively, which is shown in the Table No 7(c).

Table No 7(c)

                                Higher numbers and combining acrophpnic numerals.

One thing is to be noticed here that since there was no positional aspect of the system, so there was no need for zero as an empty place holder.

Modern Greeks still use these types of numbers for formal use, whereas we use Roman numbers, although they use Arabic numbers for calculations. However, forms of the archaic letters in particular have changed since classical Greek times.

In our modern mathematics, we use letters of various Greek alphabets to represent numbers as well, but in a completely different way. In Algebra, we use letters to represent unknowns, or variables in a general formula.

In ancient Greek number system, specific numbers were denoted by particular letters. In fact, modern mathematicians also use certain Greek letters to mean specific constants, but these have different values than the ones above. For instance, π (pi) is used widely now-a-days to represent the ratio of a circle’s circumference to its diameter, not the number 80, as in the old Greek system, Similarly φ represents the famous golden ratio (not 500), which has a magical role in beauties of nature. We will see its magnificence subsequently.

            References: Internet:
            Credit to:
                        Greek alphabets upper and lower cases :Free- Online-Bible Study.Org
          ( https://www.free-online-bible-study.org/greek-alphabet.html)
                      Acrophonic forms of some Greek numbers: Greek number systems (Table No.7a,7b,7c)
            Prof. E.F.Robertson
(http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html)

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