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THE MYSTERY OF THE DISCOVERY OF ZERO.

The idea of zero seems like such a simple and obvious concept to us that we often take its discovery for granted. Many assume that it was part of the sophisticated mathematics of the Greeks, the originators of formal geometry and logic. We are taught about zero in primary school, geometry in secondary school, and logic in university. Therefore, many people believe logic and geometry are mathematically more sophisticated than the concept of zero. (1) In actuality, the Greeks never developed the operational notion of zero, even though their achievement in geometry and logic was unparalleled. Their mathematics was completely devoid of this concept. As a consequence, their arithmetic calculations were laborious and their development of algebra was stunted because they lacked a symbolic notation.

Zero was an invention of the Hindu mathematicians, working more than 2000 years ago. Their discovery of zero led them to positional numbers, simpler arithmetic calculations, negative numbers, algebra with a symbolic notation, the idea of infinitesimals, infinity, fractions and irrational numbers. It has always been a source of mystery and surprise to the historians of mathematics that the germinal idea of zero was a discovery of Hindus and not the Greeks. The great mathematician of the eighteenth century, Laplace, wrote:
It is India that gave us the ingenious method of expressing all numbers
by means of ten symbols, each symbol receiving a value of position as
well as an absolute value, a profound and important idea which appears
so simple to us now that we ignore its true merit. But its very
simplicity and the great ease which it has lent to all computations put
our arithmetic into the first rank of useful inventions, and we shall
appreciate the grandeur of this achievement the more when we remember
that it escaped the genius of Archimedes and Appolonius, two of the
great men produced by antiquity. (2)


More recently, Tobias Dantzig, discussing the discovery of zero and positional numeration, wrote:
One who reflects upon the history of reckoning up to the invention of
the principle of position is struck by the poverty of achievement. This
long period of nearly five thousand years saw the fall and rise of many
a civilization, each leaving behind it a heritage of literature, art,
philosophy and religion. But what was the next achievement in the field
of reckoning, the earliest art practiced by man? An inflexible
numeration so crude as to make progress well nigh impossible, and a
calculating device so limited in scope that even elementary
calculations called for the services of an expert....
When viewed in this light, the achievement of the unknown Hindu who
some time in the first century of our era discovered the principle of
position assumes the proportions of a world event. Not only did this
principle constitute a radical departure in method, but we know now
that without it no progress in arithmetic was possible....
Particularly puzzling to us is the fact that the great mathematicians
of Classical Greece did not stumble upon it. Is it that the Greeks had
such a marked contempt for applied science, leaving even the
instruction of their children to slaves? But if so, how is it that the
nation that gave us geometry and carried this science so far did not
create even a rudimentary algebra? Is it not equally strange that
algebra, that cornerstone of modern mathematics, also originated in
India and at about the same time when positional numeration did? (3)


Pointing out that the discovery of zero escaped Pythagoras, Euclid and Archimedes, Constance Reid, in From Zero to Infinity, muses, "For the great mystery of zero is that it escaped even the Greeks." (4)

I shall attempt in this essay to explain the mystery posed by Reid and Dantzig: Why were the ideas of zero and algebra developed in India and not ancient Greece? I believe that the explanation of this phenomenon does not lie in an examination of Greek mathematics but rather, in an examination of Greek philosophy and logic and its contrast with Hindu philosophy and religious thought. Paradoxically, the position I reach is that the rational and logical thought patterns of the Greeks hindered their development of algebra and the invention of zero.

The Inhibition of Greek Imagination Due to Logical Rigour

To the Ancient Greeks we owe the invention and development of formal logic. They were the most logically rigorous thinkers of their time. Their logical rigour has only been surpassed since the Renaissance by modern day logicians and mathematicians who based their work on the foundations laid by the Greeks. Logical arguments permeated all areas of Greek scientific and humanistic thinking. Plato's dictum, "Let him not enter who knows not geometry," which hung over the entrance to his Academy, reflects the high value the Greeks placed on analytic thought. Perhaps the greatest contribution of Greek thought was the development of rational analytic techniques and their application to so many fields of human knowledge. As has happened so frequently in the history of ideas, Greek thinkers became captives of their new tools. Greek thinkers, using logical arguments, rejected the validity of empirically observable phenomena such as motion and change. Certain theoretical ideas such as infinity, infinitesimals, atoms, and the vacuum were also rejected by the mainstream of Greek thought because of considerations of logic. The Greeks in a sense became slaves to the linear either-or orientation of their logic, and this claim limited their imagination--making it impossible for them to conceive of the concept of zero.

The Hindus, on the other hand, had a much more relaxed intellectual tradition as far as formal logic was concerned. This, as we shall see, proved among other things an invaluable asset to Hindu mathematicians, and allowed the development of zero. The history of zero actually begins with the Babylonians, long before the Greeks or the Hindus ever engaged in their respective mathematical studies. The Babylonians used a very primitive notion of zero, which, for one reason or another, they were never able to develop as the Hindus subsequently did. The Babylonian scribes used a symbol to denote a blank space which was used as a place holder. They were on the verge of developing a place number system, but for some unknown reason they were never able to go beyond this primitive stage. They did not use their zero, for example, to distinguish between 60,1 and 1/60 in their sexagesimal base (60) number system. There is also no hint that they ever thought of zero as a number to be added or subtracted or to be used to simplify their calculations.
Babylonian positional notations also had disadvantages. It is true that
in practice the lack of distinction between the symbol for 1 and 60 is
not serious because the order of magnitude is usually known from
context; but in theoretical problems it can be very unpleasant. Still
more so is the fact that the notation does not enable us to distinguish
between 1,0,30 and 1,30 because there is no cipher. To overcome this
drawback a separate sign was introduced later on for the empty place
between two digits. (5)


An interesting footnote to the history of mathematics and our story of zero is the totally independent development in the New World of zero by the Mayans, long after its invention by the Hindus. The Mayans, like the Babylonians, only used their zero for place numeration. The Mayan zero was only used for counting and never entered into arithmetic calculations such as multiplication or division. Like the Babylonian zero, the Mayan zero was also a dead end.

Another interesting characteristic of the Babylonian and the Mayan number system is that they were both based on astronomical considerations. The Babylonian system used base 60. The Mayan system was a mixture of the base 20 and 18. The Mayan year of 360 days was divided into 18 months of 20 days each. Their numbers were used to represent the number of days between astronomical events. They used numbers up to three digits. The first digit represented the number of days and ran from 0 to 19. The second digit represented the number of 20 day periods or months. This digit varied from 1 to 17. The third digit represented the number of 18 x 20, i.e., 360, day periods or years. The number 532 represented 5 years, 3 months and two days, or 5 x (18 x 20) + 3 x 20 + 2 days, or 1,1862 days. The number 1 [??], where [??] is the Mayan zero, represents 1 x 20 + 0 days, or 20 days. The numbers 2 [??] and 1 [??] [??] would represent 40 and 360 in this system.

Returning to the question of the development of zero in the Old World, we might ask ourselves whether or not the Hindu invention of zero was independent of the Babylonians. It is possible that the Hindus could have borrowed the idea because of the commerce between Mesopotamia and India and the fact that zero appears as early as the 2nd century B.C. in India. The likelihood of finding any historic evidence to decide this question one way or another is quite small. Even if it were true that the Hindus borrowed the idea from the Babylonians, one is still left with the mystery of why the Greeks did not borrow the Babylonians' primitive notion of zero and develop it in the manner in which the Hindus did. The ancient Greeks were, in fact, closer to the Babylonians in time and space and hence would have been in a better position to borrow than the Hindus. The transmission from the Babylonian to the Greeks during the Seleucid period of the third century B.C. takes place a full century before the first appearance of zero in the Hindu literature. Neugebauer indicates that Babylonian (Mesopotamian) material was available to both traditions:
All that we can safely say is that a continuous tradition must have
existed, connecting Mesopotamian mathematics of the Hellenistic period
with contemporary Semitic (Aramic) and Greek writers and finally with
the Hindu and Islamic mathematicians. (6)


Whether or not the original primitive notion of zero was borrowed, we are still left with the mystery: Why were the Hindus able to deal with zero, and not the Greeks?

In order to gain an insight into "the great mystery" of why zero "escaped" the mathematically sophisticated Greeks, let us study the general nature of Greek thought. By examining the unique foundation of Greek philosophic, moral, scientific and mathematical thought, we shall discover the inhibitory factors which prevented the Greeks from developing the notion of zero.

Soon after the introduction of alphabetic writing into Greece, a unique transformation of her intellectual life began. (7) Beginning with Thales, the Greeks developed the notion of deductive logic and rationality. Thales began this tradition with his geometry and his science. He was the first to formally prove the empirically discovered theorems of geometry of the Egyptians and Babylonians. In physics he developed the notion that the universe was guided by a single principle, water. His student, Anaximander, chose as his all-embracing principle the neutral substance, apieron (the boundless), out of which the opposites - such as the hot and the cold or the wet and the dry emerged. Anaximander's system reflected the yes-no, either-or orientation of Greek logic.

The same orientation is reflected in the work of Heraclitus. Heraclitus developed a world system in which the universe is ruled by Logos, or Reason. The term logos derives from the Greek idea of ratio or measure. The dominance of Logos in Heraclitus' universe is a reflection of the extent to which the logical and rational mode had invaded all areas of Greek thinking. All thought, whether social, ethical, political, scientific or mathematical, was subjected to logical rigour. This new intellectual tool became the be-all and end-all of Greek thought. It enabled the Greeks to reach great intellectual heights, but it also hamstrung them by restricting their creativity to the narrow confines of logical rigour.

The first Greek philosopher to use logic in a self-consistent way was Parmenides, who, reacting to Heraclitus' notion of constant flux, wanted to prove, using logic, that nothing changes. Parmenides argued that non-being could not be because it was a logical impossibility. He then went on to argue that nothing changes because if state A were to change into state B, then state A would not be. But since non-being is impossible, state A cannot not-be, and hence cannot change into state B; hence nothing changes. This argument, naive by today's standards, had an enormous impact on Greek philosophy. All Greek philosophers who followed Parmenides incorporated his idea of non-existence from non-Being and change in their philosophic system--from Democritus' and Leucippus' unchanging atoms and Empedocles' unchanging elements (earth, air, fire and water) to Plato's division of the changing and impermanent World of Perceptions and the unchanging and permanent World of Ideas and Aristotle's distinction between the changing sub-lunar world and the unchanging ethereal heavens.

Parmenides' idea that nothing changes was incorporated into all facets of Greek thought. The idea of non-Being was rejected. Is it any wonder that the Greeks had trouble formulating the idea of zero? Once they logically connected their idea of Being, there was no longer a resonant interval. Once the interval of "play" (as between wheel and axle) was removed by logical rigour and connectedness, there was no more possibility of zero. This attitude is reflected in Aristotle's famous adage, "Nature abhors a vacuum," which formed one of the basic tenets of his physics. The climate of Greek thought was as unfavourable as possible to the formulation of zero, particularly zero as something to be manipulated mathematically as a number. The Indians, on the other hand, were used to dealing with the notion of non-Being:
In Buddhism, negativity and non-being are positive and good because the
Buddhist takes his point of departure in the negative side of life and
the world. For him the being of existence is a "nothing"; likewise
nonbeing is the negation of something negative and is, therefore,
something positive. The Greeks and the Hebrews were united in the idea
that nonbeing is something dreadful; being, however, is a genuine
reality and the true good, regardless of whether being is thought of as
eternally resting conforming to the Greek kind or in eternal motion
conforming to the Hebrew kind. (8)


For both the Hindu and the Buddhist, the notion of non-Being was a state that they actively sought in their attempt to achieve Nirvana, or oneness with the whole cosmos. Non-Being was something--a state that could be discussed. The concept of zero as a concrete state was totally consistent with this aspect of Hindu philosophy, and hence presented no problems to Hindu mathematicians. The Hindus did not have any logical stumbling blocks to overcome, like the Greeks. Nothing stood in the way of their formulation of zero. In fact, their religious beliefs encouraged this development. In conclusion, my hypothesis is that Greek thought discouraged the conceptualization of zero, whereas Hindu thought encouraged this notion.

This explanation of why the Hindus, and not the Greeks, invented zero solves only one mystery. There remains the question of why it was that the Hindus also developed the notions of 1) negative numbers, 2) simple arithmetic computational methods (algorithms), 3) algebra, with a symbolic notation, 4) infinitesimals, and 5) infinity. While the Greeks touched on all these topics, they never developed them as fully as the Hindus. They never developed a notation for the unknown quantity in their algebra. Dantzig suggests that the Greeks were at a disadvantage in developing a symbolic notation because they used their letters to denote numbers and hence had no symbols available for representing unknown variables. (9) This explanation does not strike at the heart of the matter. The Greeks could have invented new symbols, as they did when they transformed the Phoenician alphabet into the Greek alphabet and invented new letters. The Hindus did not originally use letters to denote the unknown quantity in their equations. They first used their symbol for zero and later the abbreviations of the colours. The explanation for the Hindu development of algebra and the other ideas in the preceding list, and for the Greek failure to do so, is, simply, that the Hindus pioneered the concept of zero and the Greeks did not. Each of the ideas in the preceding list is somehow related to the concept of zero. In order to see this connection, let us trace the development of zero in Hindu mathematics, and its consequences.

First Appearance of Zero
The zero symbol was used in metrics by Pingala (before 200 B.C.) in his
Chandah Sutra. (10)


The use of zero in calculations appears in the Bakhshali manuscript (200 A.D.) where use of place value is also found. The treatment of zero as a number, with equal status to other numbers such as one or two, is found in the Pancasiddhantika:
In Aries the minutes are seven, in the last sign six; in Taurus six
(repeated thrice; five (repeated) twice; four; in Gemini they are
three, two, one, zero (sunya) each repeated twice. (11)


The name for zero used in this text and later texts is sunya (pronounced shunya), which literally means "empty space" or "blank." Zero was first symbolized as dots in the Bakhshali manuscript, and later as small circles or o. The use of dots to represent zero is used as metaphor in the Vasavadatta of Subandhu:
And at the time of the rising of the moon with its blackness of night,
bowing low, as it were, with folded hands under the guise of closing
blue lotuses, immediately the stars shone forth... like sunya-bindu
(zero-dots), because of the nullity of metempsychosis, scattered in the
sky as if on the ink-blue skin rug of the Creator who reckoneth the sum
total with a bit of the moon for chalk. (12)


The operations of addition and subtraction with zero first appear in 505 A.D., in the Pancasiddhantika of Varahamihira. Brahmagupta, in 628 A.D., contains a definition of zero as a - a = o. (13) By 750 A.D., zero has all the algebraic properties it now possesses. In the Trisatika, Sridhara writes:
In addition cipher makes the sum equal to the additive, when cipher is
subtracted (from a number) there is no change in the number. On
multiplication and other operations (division by another number,
squaring, square rooting, cubing and cube rooting) on zero the result
is zero. Multiplication of a number by cipher also gives zero. (14)


Place Numeration

The most important application of zero was the development of place numeration, whereby all numbers can be represented by ten symbols-1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Our present number system, which was invented by the Hindus, was transmitted to Europe by Arab and Persian scholars. The mathematicians of Baghdad around 1000 A.D. adopted the Hindu system. They translated the Hindu term sunya into the Arabic sifr, which also means "empty space."

Originally, cipher was intended to denote only the zero element, the unique element of the number system, but eventually the entire number system became known as the cipher system. As with other innovations, the new number system was forbidden in some cities and was used secretly. The word cipher soon came to mean code, as indicated by the word decipher. In order to distinguish between the entire number system and the unique element zero, the term "zero," short for zepharino (the Latinized form of cipher), came into use to denote the zero element.

Place numeration not only simplified the expression for numbers; it made possible a revolution in the simplification of arithmetic calculations. The present-day procedures of multiplying, dividing, square and cube rooting were all developed using the place number system invented by the Hindus. Many of the Hindu techniques were transmitted to Europe by an Arab mathematician named Al Kworismi, from whose name we derive the term algorithm, meaning a procedure for performing a mathematical calculation. The scientific revolution of the Renaissance would never have taken place if these simpler modes of calculation were not made possible by the Hindu place number system.

The scientific revolution was also made possible by the Hindu development of algebra. The Hindu successes in developing algebra stemmed, like their success with zero, from their ability to work intuitively without being held back by the need for logical rigour. Part of their success may also be attributed to the fact that they had already developed zero, which proved to be a powerful concept for developing their mathematics further.

Algebra, or Avyakat-Ganita

The first place in Hindu literature where algebra or avyakat-ganita (literally, the science of calculation with the unknown) appears is in the Bakhshali manuscript (200 A.D.), one of the first works in which the concept of zero appears. It is here that we first find the concept of negative numbers. (15) The number -7 is denoted by 7 +. It is believed that the + sign represents the letter kha, the first letter of the Hindu word Ksaya, which means "to diminish." In later work, -7 is denoted by 7 or [??] where * or 0 stands for zero. This shows explicitly the relation of zero and negative numbers, which is also contained in the definition of zero:

a - a = 0.

It is also in the Bakhshali manuscript that the first representation for the unknown is found. The expression used was yadrccha venyase sunya, which literally means "put a zero (sunya) in the place of the unknown or desired quantity." (16) The symbol 0 was also used to represent the unknown. Sometimes an elongated version of this symbol, [??], was used to distinguish it from zero.

A typical example of the use of 0 for the unknown is found in the Trisatika of Sridhara (750 A.D.). (17) An arithmetic series whose first term (adih) is 20, whose number of terms (gaccah) is 7, whose sum (ganitum) is 245 and whose common difference (uttarah) is unknown, is represented by:

|adih 20|u 0|gaccah 7|ganitum 245|

The common difference is five. In modern notation we would represent the problem above as:

20 + (20 + X) + (20 + 2X) + (20 + 3X) + (20 + 4X) + (20 + 5X) + (20 + 6X) = 245 140 + 21X=245 21X = 105 X = 5

The use of sunya (zero) to denote the unknown incorporated the idea of leaving a space blank for the unknown. This idea started the Hindu mathematicians off along the road which lead to the development of algebra. The use of the symbol for zero to denote the unknown suffers, however, certain limitations. For one thing, there is the possibility of confusing the unknown with zero. It is sometimes useful to incorporate zero into an algebraic equation, which would be difficult if the unknown is also represented by zero. Another limitation is that sometimes more than one unknown appears in an equation. Once the Hindus had developed the notion of dealing with equations by incorporating an unknown through the use of sunya or zero, they expanded their notation. One of their tricks was to use the abbreviations of colours to represent multiple unknowns. An example of this is found in the representation of the equation 197X - 1644y - Z = 6302, in the Prthudakasvami of 860 A.D., as:

[mathematical expression not reproducible]

The two lines are equated to each other. The symbols yd, ka and ni are abbreviations for yavaka (red), kalaka (black) and nilaka (blue), the unknowns. The symbol ru means "to this add." The term "ka 1644" means "- 1644 X black" and "ka 0" means "no black."

The development of modern mathematics in the Renaissance picks up basically where the Hindus left off in the development of algebra. The Arabic literature served to transmit the ideas of the Hindus to the Europeans. In fact, the term "algebra" comes from the title of Al Kworismi's book, "Algebar wal Muquabalah," which literally means "On Restitution and Adjustment."

The Infinite and the Infinitesimal

Although the mathematics of infinitesimals and infinities was principally developed by the mathematicians of the Renaissance, these two concepts were first mathematically formulated by the Hindus. The concept of infinity arose through the consideration of the division of a finite number by zero. In the Bhaskara II manuscript of 1150 A.D., the quotient of the division of a finite number by zero is called Khakara.
In this quantity consisting of that which has cipher for its division,
there is no alteration, though many may be inserted or extracted as no
change takes place in the infinite and immutable God, at the period of
the destruction or creation of worlds, though numerous orders of being
are absorbed or put forth. (19)


This image of infinity is used again by Bhaskara to illustrate the idea that [infinity] + K = [infinity]:
At the time of the world's creation, the Infinite and Indestructible
Lord Almighty creates crores (1 crore = [10.sup.7]) of beings. At the
time of the great Deluge, all these beings go back to His form and are
immersed in Him. Neither process makes any change in Him. (20)


The Greeks could never have tolerated any notions like this. To their way of thinking, statements of this nature were irrational and hence, meaningless. They denied the concept of infinity out of hand as logically impossible. This attitude is described by Dantzig as the Greeks' "horror infiniti": "The infinite was taboo, it had to be kept out, at any cost; or, failing this, camouflaged by arguments ad absurdum and the like." (21)

Once one possesses the notion of infinity, the idea of an infinitesimal follows rather naturally. An infinitesimal is the result of dividing and sub-dividing a finite quantity an infinite number of times. Evidence for the existence of the notion of an infinitesimal in Hindu mathematics is found in the Bhaskara II manuscript, where one finds the statement:
The product of (a number and) zero is zero, but the number must be
returned as a multiple of zero if any further operations impend. (22)


One also finds the example [a/o] X o = a, which is correct if 0 is an infinitesimal, [epsilon]. (23)

The Greeks, although they came close to the notion of infinitesimals through the paradoxes of Zeno, Eudox's method of exhaustion and Archimedes' determination of [pi] by calculating the area of successively more-sided inscribed and circumscribed polygons to a circle, never developed the concept. The Greek "horror infiniti" deterred any effort to develop the idea of the infinitesimals at this early stage in the history of mathematics.

Fractions and Irrational Numbers

We have seen that the Greeks' logical rigour accounts for the difference in attitude between them and the Hindus towards zero, infinity, algebra, negative numbers, etc. This difference in temperaments also reflects itself in their respective attitudes towards fractions and irrational numbers.
Official Greek mathematics before Archimedes does not have any
fractions at all. This was not however because they were not known, but
rather because one did not wish to know them. For according to Plato,
the unit was indivisible and, in Plato's own words, "the experts in
this study" were absolutely opposed to dividing the unit. (The Republic
525E) Fractions were scorned at, left to the merchants; for, so it was
said, visible things are divisible, but not mathematical units. Instead
of operating with fractions, they operated with ratios of integers. (24)
The Hindus, on the other hand, had no ideo-"logical" aversion to
dealing with fractions.
The rational operations with integers and fractions, as they are taught
in our schools, are formed in exactly the same forms in the Hindu
arithmetic books. With the numbers themselves, they reached us by way
of the Arabs. (25)


The Greeks also found the notion of irrational numbers totally abhorrent. There is a legend that a member of a Pythagorian society was thrown overboard on a voyage for revealing to a member of the crew the darkly held secret that the length of the hypotenuse of a right isosceles triangle whose sides were each of unit length could not be expressed as the ratio of two integers. This fact, which was so disturbing to the Greek mathematical mind, posed no problem to the Hindus.
It should be mentioned also that the Hindus, unlike the Greeks,
regarded irrational roots of numbers as numbers. This was of enormous
help in algebra, and Indian mathematicians have been much praised for
taking this step; but one must remember that the Hindu contribution in
this case was the result of logical ignorance rather than of
mathematical insight. (26)


In conclusion, we have considered a number of examples of how the logical rigour of Greek thought hampered their mathematical development. Ideas such as zero, place numbers, fractions, negative numbers, irrational numbers, infinity, infinitesimals, and a symbolic notation for algebra developed more naturally with the Hindus, who were logically less sophisticated than the Greeks, but were, nevertheless, capable of powerful intuitive insights. These Hindu discoveries have played a key role in the subsequent development of Western science and mathematics.

Notes and References

(1.) This is another example of the distortion of the history of science effected by textbooks and school curricula, as pointed out by Thomas Kuhn in The Structure of Scientific Revolutions.

(2.) This quote can be found in T. Dantzig, Number, The Language of Science, 4th Edition (1954), p. 19-20.

(3.) T. Dantzig, op. cit., p. 30.

(4.) C. Reid, From Zero to Infinity (1964), p. 4.

(5.) B. Van der Waerden, Science Awakening (1954), p. 39.

(6.) O. Neugebauer, The Exact Science in Antiquity (1952), p. 141.

(7.) This process has been described by H.M. McLuhan in The Gutenberg Galaxy (1962), and by E. Havelock in Preface to Plato (1963).

(8.) T. Boman, Hebrew Thought Compared with Greek (1960), p. 57-8.

(9.) T. Dantzig, op. cit., p. 80.

(10.) B. Datta and A. N. Singh, History of Hindu Mathematics, A Source Book (1962), Part I, p. 75-81.

(11.) Ibid.

(12.) Ibid.

(13.) Ibid., Part I, p. 239.

(14.) Ibid., Part I, p. 240.

(15.) C. N. Srinivasiengar, The History of Ancient Indian Mathematics (1967), p. 32.

(16.) Ibid.

(17.) Ibid.

(18.) B. Datta and A.N. Singh, op. cit, Part II, p. 31.

(19.) Ibid., Part I, p. 243.

(20.) C.N. Srinivasiengar, op. cit., p. 82.

(21.) T. Dantzig, op. cit, p. 129-30.

(22.) B. Datta and A.N. Singh, op. cit, Part I, p. 242.

(23.) Ibid., Part I, p. 243.

(24.) B. Van der Waerden, op. cit, p. 49.

(25.) Ibid., p. 57.

(26.) C. Bayer, A History of Mathematics (1968), p. 242.

ROBERT K. LOGAN (*)

(*) Robert Logan is an Associate Professor of Physics in the Department of Physics at the University of Toronto.
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