It has become an accepted convention that science began in ancient Greece by one of the reputed sages called Thales, who lived in the 7-6th century BC.
But we come across the early texts of India certain trends of thought that are scientifically significant and to a person – Uddalaka Aruni of the Gautama clan – as having been the initiator of this new direction of systematically investigating nature. The science intoxicated Uddalaka must have been earlier than Thales whose actual teachings moreover give us the impression of having been far more profound from the viewpoint of science-potential than all that we know about Thales.
But historians of science remain unaware of the very name of Uddalaka Aruni. Internally, in his own country, his views are subjected to almost endless distortions, energetic efforts being made for centuries to make him appear as a religion-oriented extreme idealist philosopher. Externally, most of the historians of science have so far worked under the spell of what is often described as Euro-centrism – that science is an essentially European phenomenon.
J.D. Bernal goes to the extent of using the rather exasperated expression “arrogant ignorance” as forming the main prop of this Euro-centrism.
With pronounced bias for Euro-centrism, the otherwise admirable French historian, Arnold Reymond, claims that nature science owes its origin to the peculiar genius of the Greeks, or more simply to some kind of “Greek miracle”. Who then is the miracle maker and what were his achievements? Since Thales himself leaves for us nothing in writing and since we are confronted with all sorts of floating legends about him – one example, wanting us to believe that as an idle star gazer he fell into a well, another insisting on his practical wisdom enabling him to earn a lot in olive business.
India (who never invaded any country in her last 10000 years of history) invented the Number System; even the zero, was invented by Aryabhatta. An important Mathematics book prescribed by the New York State Education Department acknowledges the debt in the following words: «The Western world owes a great deal to India for a simple invention. It was developed by an unknown Indian more than 1500 years ago. Without most of the great discoveries and inventions (including computers) of western civilization would never have come about. This invention was the decimal system of numerals — nine digits and a zero. The science and technology of today (including the computers) could not have developed if we had only the Roman system of numerals. That system is too clumsy to be used as a scientific too. Today we take the decimal system for granted. We don’t think about how brilliant the man who invented zero must have been. Yet without zero we could not assign a place value to the digits. That ancient mathematician, whoever, he was, deserves much honour. Indians also made advances in other areas of mathematics. Very early in their history they developed a simple system of geometry. This system was used to plan outdoor sites for religious ceremonies. Indians also added to our knowledge of even more complicated branches of mathematics such as trigonometry and calculus. They studied these branches of mathematics in order to apply them to astronomy».
The World’s first university was established in Takshila in 700 BC. More than 10,500 students from all over the world studied more than 60 subjects. The University of Nalanda built in the 4th century CE was one of the greatest achievements of ancient India in the field of education.
Sanskrit, the mother of all the european languages, is the most suitable language for the computer software — as report the Forbes magazine (July 1987).
Ayurveda is the earliest school of medicine known to humans. Charaka, the father of medicine consolidated Ayurveda 2500 years ago. Today Ayurveda is fast regaining its rightful place in our civilization.
Although modern images of India often show poverty and lack of development, India was the richest country on earth until the time of British in the early 17th Century. Christopher Columbus was attracted by her wealth.
Even the art of Navigation was born in the river Sindh 6000 years ago. The very word ‘navigation’ is derived from the Sanskrit word nav gatih. The word navy is also derived from Sanskrit nou.
Bhaskaracarya calculated the time taken by the earth to orbit the sun hundreds of years before the astronomer Smart. Time taken by earth to orbit the sun: (5th century) 365,258756484 days.
The value of ‘pi’ was first calculated by Budhayana, and he explained the concept of what is known as the Pythagorean Theorem. He discovered this in the 6th century long before the European mathematicians. Algebra, Trigonometry and Calculus came from India. Quadratic equations were propounded by Sridharacarya in the 11th century. The largest numbers the Greeks and the Romans used were 106 whereas hindus used numbers as big as 1053 with specific names as early as 5000 BCE during the Vedic period.
The earliest reservoir and dam for irrigation was built in Saurashtra. According to Saka King Rudradaman I of 150 CE a beautiful lake aptly called ‘Sudarshana’ was constructed on the hills of Raivataka during Chandragupta Maurya’s time.
Chess (Shataranja or Ashta Pada) was invented in India.
Sushruta is the father of surgery. 2600 years ago he and health scientists of his time conducted complicated surgeries like cesareans, contracts, artificial legs, fractures, urinary stones and even plastic surgery and brain surgery. Usage of anesthesia was well known in ancient India. Over 125 surgical equipment’s were used. Deep knowledge of anatomy physiology, etiology, embryology, digestion, metabolism, genetics and immunity is also found in many texts.
When Europeans were only nomadic forest dwellers over 5000 years ago, indians established Harappan culture in Sindhu Valley (Indus Valley Civilization) The place value system, the decimal system was developed in India in 100 BC. All of us who wear cotton cloth, do yoga, seek peace of mind or tranquillity through meditation, are indebted to India.
The forging of wrought iron seems to have reached its zenith in India in the first millennium AD. The earliest large forging is the iron pillar at New Delhi dated by inscription to the Gupta period of the 3rd century AD at a height of over 7 m and weight of about 6 tons. The pillar is believed to have been made by forging together a series of disc-shaped iron blooms. Apart from the dimensions another remarkable aspect of the iron pillar is the absence of corrosion which has been linked to the composition, the high purity of the wrought iron and the phosphorus content and the distribution of slag. Town Planning and Great Baths of Indus Valley Urban planning is evident in the neat arrangement of the major buildings contained in the citadel, including the placement of a large granary and water tank or bath at right angles to one another. The lower city, which was tightly packed with residential units, was also constructed on a grid pattern consisting of a number of blocks separated by major cross streets. Baked-brick houses faced the street, and domestic life was centered around an enclosed courtyard.
The cities had an elaborate public drainage system, Sanitation was provided through an extensive system of covered drains running the length of the main streets and connected by chutes with most residences.
The statue of Nataraja (dance pose of Lord Shiva) is a well known example for the artistic, scientific and philosophical significance. The late scientist Carl Sagan, asserts that the dance of Nataraja signifies the cycle of evolution and destruction of the cosmic universe (Big Bang Theory). «It is the clearest image of the activity of God which any art or religion can boast of». Modern physics has shown that the rhythm of creation and destruction is not only manifest in the turn of the seasons and in the birth and death of all living creatures, but also the very essence of inorganic matter. For modern physicists, then, Shiva’s dance is the dance of subatomic matter. Hundreds of years ago, Indian artist created visual images of dancing Shivas in a beautiful series of bronzes. Today, physicists have used the most advanced technology to portray the pattern of the cosmic dance. Thus, the metaphor of the cosmic dance unifies, ancient mystic art and modern physics.
Ideas of natural selection, atomic polarity and evolution. This is what Manu said, perhaps 10,000 years before the birth of Christ: the first germ of life was developed by water and heat.
In the valley of the Indus River of India, the world’s oldest civilization had developed its own system of mathematics. The Vedic Shulba Sûtra (fifth to eighth century BCE), meaning “codes of the rope”, show that the earliest geometrical and mathematical investigations among the Indians arose from certain requirements of their rituals. When the poetic vision of the vedic seers was externalized in symbols, rituals requiring altars and precise measurement became manifest, providing a means to the attainment of the unmanifest world of consciousness. Shulba Sûtra is the name given to those portions or supplements of the Kalpasûtra, which deal with the measurement and construction of the different altars or arenas for rites. The word shulba refers to the ropes used to make these measurements. The word sulva is derived for the root sulv, to measure. Since the cord or rope, rajju was used for measuring, in course of time sulva became a synonym for rope. The date of composition of the earliest these books must be much before 1800 BCE when the Sarasvatî river dried up and the Vedic civilisation was on the decline. Here we give only a brief overview of the four Sulba Sûtra books associated with the names of Baudhayana, Apastamba, Katyayana and Manava. Although vedic mathematicians are known primarily for their computational genius in arithmetic and algebra, the basis and inspiration for the whole of Indian mathematics is geometry. Evidence of geometrical drawing instruments from as early as 2500 BCE has been found in the Indus Valley. The beginnings of algebra can be traced to the constructional geometry of the vedic hierophants, which are preserved in the Shulba Sûtra. Exact measurements, orientations, and different geometrical shapes for the altars and arenas used for the ritualistic functions (yajña), which occupy an important part of the vedic culture, are described in the Shulba Sûtra. Many of these calculations employ the geometrical formula known as the Pythagorean theorem.
This theorem (c. 540 BCE), equating the square of the hypotenuse of a right angle triangle with the sum of the squares of the other two sides, was utilized in the earliest Shulba Sûtra (the Baudhayana) prior to the eighth century BCE. Thus, widespread use of this famous mathematical theorem in India several centuries before its being popularized by Pythagoras has been documented. The exact wording of the theorem as presented in the Shulba Sûtra is: «The diagonal chord of the rectangle makes both the squares that the horizontal and vertical sides make separately». The proof of this fundamentally important theorem is well known from Euclid’s time until the present for its excessively tedious and cumbersome nature; yet the Veda present five different extremely simple proofs for this theorem. One historian, Needham, has stated, «Future research on the history of science and technology in Asia will in fact reveal that the achievements of these peoples contribute far more in all pre-Renaissance periods to the development of world science than has yet been realized».
The Shulba Sûtra have preserved only that part of Vedic mathematics which was used for constructing the altars and for computing the calendar to regulate the performance of religious rituals. After the Shulba Sûtraperiod, the main developments in Vedic mathematics arose from needs in the field of astronomy. The Jyotisha, science of the luminaries, utilizes all branches of mathematics. The need to determine the right time for their rituals gave the first impetus for astronomical observations. With this desire in mind, the attendants would spend night after night watching the advance of the moon through the circle of the nakshatra (lunar mansions), and day after day the alternate progress of the sun towards the north and the south. However, the officiants were interested in mathematical rules only as far as they were of practical use. These truths were therefore expressed in the simplest and most practical manner. Elaborate proofs were not presented, nor were they desired.
A close investigation of the Vedic system of mathematics shows that it was much more advanced than the mathematical systems of the civilizations of the Nile or the Euphrates. The Vedic mathematicians had developed the decimal system of tens, hundreds, thousands, etc., where the remainder from one column of numbers is carried over to the next. The advantage of this system of nine number signs and a zero is that it allows for calculations to be easily made. Further, it has been said that the introduction of zero, or shunya as the Indians called it, in an operational sense as a definite part of a number system, marks one of the most important developments in the entire history of mathematics. The earliest preserved examples of the number system which is still in use today are found on several stone columns erected in India by King Ashoka in about 250 BCE. Similar inscriptions are found in caves near Poona (100 BCE) and Nasik (200 CE). These earliest Indian numerals appear in a script called brahmî.
After 700 CE another notation, called by the name “Indian numerals”, which is said to have evolved from the brahmî numerals, assumed common usage, spreading to Arabia and from there around the world. When Arabic numerals (the name they had then become known by) came into common use throughout the Arabian empire, which extended from India to Spain, Europeans called them “Arabic notations”, because they received them from the Arabians. However, the Arabians themselves called them “Indian figures” (Al-Arqan-Al-Hindu) and mathematics itself was called “the Indian art” (hindisat).
Mastery of this new mathematics allowed the Muslim mathematicians of Baghdad to fully utilize the geometrical treatises of Euclid and Archimedes. Trigonometry flourished there along with astronomy and geography. Later in history, Carl Friedrich Gauss, the “prince of mathematics”, was said to have lamented that Archimedes in the third century BCE had failed to foresee the Indian system of numeration; how much more advanced science would have been.
Prior to these revolutionary discoveries, other world civilizations — the Egyptians, the Babylonians, the Romans, and the Chinese — all used independent symbols for each row of counting beads on the abacus, each requiring its own set of multiplication or addition tables. So cumbersome were these systems that mathematics was virtually at a standstill. The new number system from the Indus Valley led a revolution in mathematics by setting it free. By 500 CE mathematicians of India had solved problems that baffled the world’s greatest scholars of all time. Aryabhatta, an astronomer mathematician who flourished at the beginning of the 6th century, introduced sines and versed sines — a great improvement over the clumsy half-cords of Ptolemy.
A. L. Basham, foremost authority on ancient India, writes in The Wonder That Was India, «Medieval Indian mathematicians, such as Brahmagupta (seventh century), Mahavira (ninth century), and Bhaskara (twelfth century), made several discoveries which in Europe were not known until the Renaissance or later. They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations. Mahavira’s most noteworthy contribution is his treatment of fractions for the first time and his rule for dividing one fraction by another, which did not appear in Europe until the 16th century».
B.B. Datta writes: «The use of symbols-letters of the alphabet to denote unknowns, and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra». The great Indian mathematician Bhaskaracarya (1150 CE) produced extensive treatises on both plane and spherical trigonometry and algebra, and his works contain remarkable solutions of problems which were not discovered in Europe until the seventeenth and eighteenth centuries. He preceded Newton by over 500 years in the discovery of the principles of differential calculus. A.L. Basham writes further, «The mathematical implications of zero (shunya) and infinity, never more than vaguely realized by classical authorities, were fully understood in medieval India. Earlier mathematicians had taught that X/0 = X, but Bhaskara proved the contrary. He also established mathematically what had been recognized in Indian theology at least a millennium earlier: that infinity, however divided, remains infinite». In the 14th century, Madhava, isolated in South India, developed a power series for the arc tangent function, apparently without the use of calculus, allowing the calculation of ‘pi’ to any number of decimal places. Whether he accomplished this by inventing a system as good as calculus or without the aid of calculus, either way it is astonishing.
Spiritually advanced cultures were not ignorant of the principles of mathematics, but they saw no necessity to explore those principles beyond that which was helpful in the advancement of God realization.
By the fifteenth century CE use of the new mathematical concepts from India had spread all over Europe to Britain, France, Germany, and Italy, among others. A.L. Basham states also that «The debt of the Western world to India in this respect [the field of mathematics] cannot be overestimated. Most of the great discoveries and inventions of which Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was, from the world’s point of view, after the Buddha, the most important son of India. His achievement, though easily taken for granted, was the work of an analytical mind of the first order, and he deserves much more honour than he has so far received».
Unfortunately, Eurocentrism has effectively concealed from the common man the fact that we owe much in the way of mathematics to ancient India. Reflection on this may cause modern man to consider more seriously the spiritual preoccupation of ancient India. The rishi were not men lacking in practical knowledge of the world, dwelling only in the realm of imagination. They were well developed in secular knowledge, yet only insofar as they felt it was necessary within a world view in which consciousness was held as primary.
In ancient India, mathematics served as a bridge between understanding material reality and the spiritual conception. Vedic mathematics differs profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic satisfaction) did not appeal to the Indian mind. The mathematics of the Veda lacks the cold, clear, geometric precision of the West; rather, it is cloaked in the poetic language which so distinguishes the East. Vedic mathematicians strongly felt that every discipline must have a purpose, and believed that the ultimate goal of life was to achieve self-realization and self-perfection. Those practices which furthered this end either directly or indirectly were practiced most rigorously. Outside of the religio-astronomical sphere, only the problems of day to day life (such as purchasing and bartering) interested the Indian mathematicians.
One of the foremost exponents of Vedic math, the late Bharati Krishna Tirtha Maharaja, author of Vedic Mathematics, has offered a glimpse into the sophistication of Vedic math. Drawing from the Atharva Veda, Tirtha Maharaja points to many sûtra or aphorisms which appear to apply to every branch of mathematics: arithmetic, algebra, geometry (plane and solid), trigonometry (plane and spherical), conics (geometrical and analytical), astronomy, calculus (differential and integral), etc.
Utilizing the techniques derived from these sûtra, calculations can be done with incredible ease and simplicity in one’s head in a fraction of the time required by modern means. Calculations normally requiring as many as a hundred steps can be done by the Vedic method in one single simple step. For instance the conversion of the fraction 1/29 to its equivalent recurring decimal notation normally involves 28 steps. Utilizing the Vedic method it can be calculated in one simple step.
In order to illustrate how secular and spiritual life were intertwined in Vedic India, Tirtha Maharaja has demonstrated that mathematical formulas and laws were often taught within the context of spiritual expression (mantra). Thus while learning spiritual lessons, one could also learn mathematical rules.
Tirtha Maharaja has pointed out that Vedic mathematicians prefer to use the devanâgarî letters of Sanskrit to represent the various numbers in their numerical notations rather than the numbers themselves, especially where large numbers are concerned. This made it much easier for the students of this math in their recording of the arguments and the appropriate conclusions.
Tirtha Maharaja states, «In order to help the pupil to memorize the material studied and assimilated, they made it a general rule of practice to write even the most technical and abstruse textbooks in sutra or in verse (which is so much easier-even for the children to memorize). And this is why we find not only theological, philosophical, medical, astronomical, and other such treatises, but even huge dictionaries in Sanskrit verse! So from this standpoint, they used verse, sutra and codes for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assimilable form)!» The Sanskrit consonants ka, ta, pa, ya all denote 1;kha, tha, pha, ra all represent 2; ga, da, ba, la all stand for 3; Gha, dha, bha, va all represent 4; jña, na, ma all represent 5; ca, ta, sa all stand for 6; and so on.
Vowels make no difference and it is left to the author to select a particular consonant or vowel at each step. This great latitude allows one to bring about additional meanings of his own choice. For example kapa, tapa, papa, yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings. Here is an actual sûtraof spiritual content, as well as secular mathematical significance:
gopi bhagya madhuvrata
sringiso dadhi sandhiga
khala jivita khatava
gala hala rasandara
While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.
The translation is as follows:
«O Lord anointed with the yogurt of the milkmaids’ worship (Krishna), O savior of the fallen, O master, please protect me».
At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0,31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in devotion, by this method one can also add to memory significant secular truths.
This is the real gist of the Vedic world view regarding the culture of knowledge: while culturing transcendental knowledge, one can also come to understand the intricacies of the phenomenal world. By the process of knowing the absolute truth, all relative truths also become known. In modern society today it is often contended that never the twain shall meet: science and religion are at odds. This erroneous conclusion is based on little understanding of either discipline. Science is the smaller circle within the larger circle of spirituality..
Albert Einstein said: «We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made».
Sir William Hunter, said India «has even contributed to modern medical science by the discovery of various chemicals and by teaching you how to reform misshapen ears and noses. Even more it has done in mathematics, for algebra, geometry, astronomy, and the triumph of modern science — mixed mathematics — were all invented in India, just so much as the ten numerals, the very cornerstone of all present civilization, were discovered in India, and are in reality, Sanskrit words.
There is considerable mathematical information, both explicit and implicit in the Veda Samhitâ. Among them, the earliest is the Rig Veda Samhita.
All the Veda Samhitâ are hymns to the various deities. However these hymns praise all forms of knowledge. There was no rigid distinction of the secular and sacred knowledge. The knowledge of mathematics, the knowledge of geometry, especially as related to the construction of houses and cities were all deemed important and worthy of mantra in the hymns. Some of these hymns dealing with the series of integers are recited even today on sacred occasions. Some of the hymns, which deal with cosmology, imply that these poets were very familiar with geometry and the planning needed to construct complex objects.
Consider for example the following verse in RV (Mandala10, Sûkta 130, Verse 3). It deals with the creation or formation of the universe.
«Who was the measurer prama? What was the model pratimâ? What were the building materials for things offered nidânam ãjyam? What is the circumference (of this universe) paridhih? What are the meters or harmonies behind the Universe chandah? What is the triangle (yoke) praugam [which connects this universe to the source of driving force, the engine]?»
All the words in Sanskrit, prama etc., are geometrical terms which also occur in the later Shulba Sûtra with the indicated meaning. Hence it is safe to assume that these poets were aware of the construction of buildings and other artifacts.
In Atharva Veda (10.2.31) the town of gods called Ayodhya is described. It is circular in plan with eight rampart walls and nine doors. Even if the poem is interpreted metaphorically, the use of a metaphor implies that poets had the experience of real things, i.e. a real physical city. Atharva Veda XIX.58.4 declares that the town should be made unconquerable using the thing called ayasa. Whether we translate it as strength or as some metal or as iron which is the word’s meaning in later times, either way it indicates that the poets were aware of complex planning of these geometrical entities.
The simplistic notion circulated by the indologists of the nineteenth century that these poets were nomads with a relatively low level of culture has absolutely no support from the hymns.
Next consider the Rig Veda Samhita which has more than 10,000 metrical couplets. Each verse has a distinct metre which imposes a structure on the verse like the total number of vowels in it and the number of vowels in each subgroup of the verse. Moreover the medium of preservation of the text was also recitation. The Vedic sages attached the greatest importance to the preservation of the text of the Rig Veda along with accent marks and developed special methods of recitation which remind us of the modern error correction and detection codes in modern communication and computer systems, i.e. what the codes do for correcting the linear printed text is done by these special methods of recitation for oral text.
Chariots are mentioned copiously in Rig Veda (I.102.3, I.53.9, I.55.7, I.141.8, II.12.8, IV.46.2,…). Chariots could also be triangular having three seats and three wheels (RV. I.118.2, I.34.2). A spoked wheel is mentioned in many places in Rig Veda. Specifically the five spoked wheel (I.164.13) and the 360-spoked wheel (I.164.48) are mentioned. The spoked wheel has four parts, hubs nâbih, fellies pradhaya, spokes shanka, âreor rim. By the time of Yajur Veda (XVI.27) the number and varieties of the manufactured chariots had increased so much that a separate guild of chariot makers was developed. Dr. Kulakarni (1983) writes: «The proficiency in chariot building presupposes a good deal of knowledge of geometry... The fixing of spokes of odd or even numbers require knowledge of dividing the area of the circle into the desired numbers of small parts of equal area, by drawing diameters. This also presupposes the knowledge of dividing a given angle into equal parts».
The Rig Veda is full of references to the words which come up in rituals, even though it does not mention any ritual in detail. The details of the rituals, especially the design of the fire altars and methods of constructing them using specially shaped bricks are given in the subsequent Brâhmana, and also, with more details, in the Shulba Sûtra. Whenever any rituals are codified in a oral or written, the implication is that they must have been in existence for much longer time. For instance consider the three types of fire altars namely garhapatya, ahavaniya and dakshina. All three are mentioned in Rig Veda. However the Shatapatha Brâhmana declares that the three fire altars are square, circular and semi circular in shape and, more importantly having the same area. All scholars such as Burk (1901), Dutta (1932), Seidenberg (1962), agree that this constraint of equal area must have been there even in the early Vedic age before the codification of Rigvedic Hymns. To construct a figure of a specific area, we need to have at least an approximate method of finding the square root of the number two. To construct a circle of area equal to that of a square, one needs to have at least an approximate value of the number ‘pi’, the ratio of circumference of a circle to the diameter. Again the books like the Shatapatha Brâhmana or the Shulba Sûtras codify the methods existing for a long time in addition to developing new methods of drawing the geometrical figures and the associated theory. The minimum knowledge needed for finding the square root of two or for drawing isosceles trapezium is that of Pythagorean triples. All this evidence implies that the Vedic sages in the early Vedic age knew some simplified versions of the Pythagorean theorem. It was there, probably that the origin of mathematics took place as argued by Seidenberg (1978) and Rajaram (1995).
The concept of infinity. The Vedic Indians were aware of the fundamental difference between a large number and infinity. They were aware that an infinite number couldn’t be produced by several finite numbers with finite number of operations.
These are many words for infinity namely ananta, purnam and aditi. The word innumerable asamkhyataoccurs in Yajur Veda XVI.54. Brihadâranyaka Upanishad (II.5.10), which is associated with Shatapata Brâhmana and Shukla Yajur Veda) in describing the count of the mysteries of Indra declares it is ananta literally meaning that which has no end or anta. They stated two clear definitions. The Atharva Veda X.8.24 states that «infinity can come out of only infinity» and «infinity is left over from infinity after operations on it». These two statements are made more precise in the invocatory verse of Isha Upanishad(Shukla Yajur Veda):
pûrnamadah pûrnamidam pûrnât pûrnamudacyate
pûrnâsya pûrnamadaya pûrnamevâvasishyate
«From infinity is born infinity.
When infinity is taken out of infinity,
only infinity is left over».
The pûrna is not limited to the mathematical infinity. The author of the hymns is trying to define the concept of all perfect perfects’. Its projection to the realm of mathematics is the mathematical infinity denoted by the symbol infinity later.