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Showing posts with label read. Show all posts
Showing posts with label read. Show all posts

Monday, August 27, 2007

pyramids

Geometry > Solid Geometry > Polyhedra > Pyramids
Recreational Mathematics > Interactive Entries > LiveGraphics3D Applets


Pyramid








A pyramid is a polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common polygon vertex (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a right pyramid whose base is a regular polygon. An -gonal regular pyramid (denoted ) having equilateral triangles as sides is possible only for , 4, 5. These correspond to the tetrahedron, square pyramid, and pentagonal pyramid, respectively.

A pyramid is self-dual, corresponding to the fact that a pyramid's skeleton (a wheel graph) is a self-dual graph.

An arbitrary pyramid has a single cross-sectional shape whose lengths scale linearly with height. Therefore, the area of a cross section scales quadratically with height, decreasing from at the base () to 0 at the apex (assumed to lie at a height ). The area at a height above the base is therefore given by

(1)

As a result, the volume of a pyramid, regardless of base shape or position of the apex relative to the base, is given by

(2)
(3)
(4)

Note that this formula also holds for the cone, elliptic cone, etc.

The volume of a pyramid whose base is a regular -sided polygon with side is therefore

(5)

Expressing in terms of the circumradius of the base gives

(6)

(Lo Bello 1988, Gearhart and Schulz 1990).

The geometric centroid is the same as for the cone, given by

(7)

The lateral surface area of a pyramid is

(8)

where is the slant height and is the base perimeter.

Joining two pyramids together at their bases gives a dipyramid, also called a bipyramid.


piyushdadriwala
www.piyushdadriwalamaths.co.in

Saturday, August 11, 2007

पी................ush

A history of Pi

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A chronology of Pi History Topics Index


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A little known verse of the Bible reads

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)

The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates: though in defence of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.

The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the 'Biblical' value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 (8/9)2 = 3.16 as a value for π.

The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation

223/71 < π < 22/7.

Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.

Here is Archimedes' argument.

Consider a circle of radius 1, in which we inscribe a regular polygon of 3 2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 2n-1 sides, with semiperimeter an.


The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing sequence

b1 , b2 , b3 , ...

and a decreasing sequence

a1 , a2 , a3 , ...

such that both sequences have limit π.

Using trigonometrical notation, we see that the two semiperimeters are given by

an = K tan(π/K), bn = K sin(π/K),

where K = 3 2n-1. Equally, we have

an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),

and it is not a difficult exercise in trigonometry to show that

(1/an + 1/bn) = 2/an+1 . . . (1)

an+1bn = (bn+1)2 . . . (2)

Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that

b6 < π < a6 .

It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.

For of course there is no reason in principle why one should not go on. Various people did, including:


Ptolemy (c. 150 AD) 3.1416

Zu Chongzhi (430-501 AD) 355/113

al-Khwarizmi (c. 800 ) 3.1416

al-Kashi (c. 1430) 14 places

Viète (1540-1603) 9 places

Roomen (1561-1615) 17 places

Van Ceulen (c. 1600) 35 places



Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes' work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.

Al-Khwarizmi lived in Baghdad, and incidentally gave his name to 'algorithm', while the words al jabr in the title of one of his books gave us the word 'algebra'. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.

The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of Wallis (1616-1703)

2/π = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...)

and one of the best-known is

π/4 = 1 - 1/3 + 1/5 - 1/7 + ....

This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).

These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.

From the point of view of the calculation of π, however, neither is of any use at all. In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series. However, Gregory also showed the more general result

tan-1 x = x - x3/3 + x5/5 - ... (-1 x 1) . . . (3)

from which the first series results if we put x = 1. So using the fact that

tan-1(1/√3) = π/6 we get

π/6 = (1/√3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...

which converges much more quickly. The 10th term is 1/(19 39√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.

An even better idea is to take the formula

π/4 = tan-1(1/2) + tan-1(1/3) . . . (4)

and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).

Clearly we shall get very rapid convergence indeed if we can find a formula something like

π/4 = tan-1(1/a) + tan-1(1/b)

with a and b large. In 1706 Machin found such a formula:

π/4 = 4 tan-1(1/5) - tan-1(1/239) . . . (5)

Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.

With a formula like this available the only difficulty in computing π is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin's formula to calculate π to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:

1699: Sharp used Gregory's result to get 71 correct digits

1701: Machin used an improvement to get 100 digits and the following used his methods:

1719: de Lagny found 112 correct digits

1789: Vega got 126 places and in 1794 got 136

1841: Rutherford calculated 152 digits and in 1853 got 440

1873: Shanks calculated 707 places of which 527 were correct



A more detailed Chronology is available.

Shanks knew that π was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that 'squaring the circle' is impossible. The transcendentality of π implies that there is no ruler and compass construction to construct a square equal in area to a given circle.

Very soon after Shanks' calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7's. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528th place, after which all his digits were wrong. In 1949 a computer was used to calculate π to 2000 places. In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.

You can see 2000 places of π.

We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states "3.14159 andc. = π". Euler adopted the symbol in 1737 and it quickly became a standard notation.

We conclude with one further statistical curiosity about the calculation of π, namely Buffon's needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π. Various people have tried to calculate π by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got

π = 355/113 = 3.1415929

which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing π. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.

Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for π was thus given by

2 0.7857 / π = 1/2

from which he got the highly creditable value of π = 3.1428. He was not being serious!

It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined π in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that π/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau's dismissal:-

Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.

G H Hardy replied immediately to Bieberbach in a published note about the consequences of this un-German definition of π

There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach's reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.

Not only in Germany did π present problems. In the USA the value of π gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.

Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)

The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!

Open questions about the number π




Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?

Brouwer's question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero?

Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?

Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?

Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.

Another normal question! We know that π is not rational so there is no point from which the digits will repeat. However, if π is normal then the first million digits 314159265358979... will occur from some point. Even if π is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.

As a postscript, here is a mnemonic for the decimal expansion of π. Each successive digit is the number of letters in the corresponding word.

How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...:

3.14159265358979323846264...


You can see more about the history of π in the History topic: Squaring the circle and you can see a Chronology of how calculations of π have developed over the years.

www.piyushdadriwalamaths.co.in

zero

A history of Zero

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One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.

Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.

Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.

Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.

The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-

... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.

In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-

... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.

However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-

A number remains unchanged when divided by zero.

Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.

Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.

Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics.

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-

If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.

The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.

Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-

An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.

In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.

One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!

An overview of Indian mathematics

An overview of Indian mathematics

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It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them.

We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.

Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed.

We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch". Of course ten units is then 13.2 inches which is quite believable as the measure of a "foot". A similar measure based on the length of a foot is present in other parts of Asia and beyond. Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is certainly surprising the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches which is the measure of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.

It is unclear exactly what caused the decline in the Harappan civilisation. Historians have suggested four possible causes: a change in climatic patterns and a consequent agricultural crisis; a climatic disaster such flooding or severe drought; disease spread by epidemic; or the invasion of Indo-Aryans peoples from the north. The favourite theory used to be the last of the four, but recent opinions favour one of the first three. What is certainly true is that eventually the Indo-Aryans peoples from the north did spread over the region. This brings us to the earliest literary record of Indian culture, the Vedas which were composed in Vedic Sanskrit, between 1500 BC and 800 BC. At first these texts, consisting of hymns, spells, and ritual observations, were transmitted orally. Later the texts became written works for use of those practicing the Vedic religion.

The next mathematics of importance on the Indian subcontinent was associated with these religious texts. 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 practical religious purposes. The mathematics contained in the these texts is studied in some detail in the separate article on the Sulbasutras.

The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). These men were both priests and scholars but they were not mathematicians in the modern sense. Although we have no information on these men other than the texts they wrote, we have included them in our biographies of mathematicians. There is another scholar, who again was not a mathematician in the usual sense, who lived around this period. That was Panini who achieved remarkable results in his studies of Sanskrit grammar. Now one might reasonably ask what Sanskrit grammar has to do with mathematics. It certainly has something to do with modern theoretical computer science, for a mathematician or computer scientist working with formal language theory will recognise just how modern some of Panini's ideas are.

Before the end of the period of the Sulbasutras, around the middle of the third century BC, the Brahmi numerals had begun to appear.




Here is one style of the Brahmi numerals..


These are the earliest numerals which, after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used today. The development of numerals and place-valued number systems are studied in the article Indian numerals.

The Vedic religion with its sacrificial rites began to wane and other religions began to replace it. One of these was Jainism, a religion and philosophy which was founded in India around the 6th century BC. Although the period after the decline of the Vedic religion up to the time of Aryabhata I around 500 AD used to be considered as a dark period in Indian mathematics, recently it has been recognised as a time when many mathematical ideas were considered. In fact Aryabhata is now thought of as summarising the mathematical developments of the Jaina as well as beginning the next phase.

The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. More surprisingly the Jaina developed a theory of the infinite containing different levels of infinity, a primitive understanding of indices, and some notion of logarithms to base 2. One of the difficult problems facing historians of mathematics is deciding on the date of the Bakhshali manuscript. If this is a work which is indeed from 400 AD, or at any rate a copy of a work which was originally written at this time, then our understanding of the achievements of Jaina mathematics will be greatly enhanced. While there is so much uncertainty over the date, a topic discussed fully in our article on the Bakhshali manuscript, then we should avoid rewriting the history of the Jaina period in the light of the mathematics contained in this remarkable document.

You can see a separate article about Jaina mathematics.

If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that "science" which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems.

Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC. If he had made a literal translation it is doubtful whether it would have been of interest to more than a few academically minded people. He popularised the text, however, by resetting the whole work into Indian culture using Hindu images with the Indian caste system integrated into his text.

By about 500 AD the classical era of Indian mathematics began with the work of Aryabhata. His work was both a summary of Jaina mathematics and the beginning of new era for astronomy and mathematics. His ideas of astronomy were truly remarkable. He replaced the two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which causes an eclipse by covering the Moon or Sun or their light, with a modern theory of eclipses. He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories.

Aryabhata headed a research centre for mathematics and astronomy at Kusumapura in the northeast of the Indian subcontinent. There a school studying his ideas grew up there but more than that, Aryabhata set the agenda for mathematical and astronomical research in India for many centuries to come. Another mathematical and astronomical centre was at Ujjain, also in the north of the Indian subcontinent, which grew up around the same time as Kusumapura. The most important of the mathematicians at this second centre was Varahamihira who also made important contributions to astronomy and trigonometry.

The main ideas of Jaina mathematics, particularly those relating to its cosmology with its passion for large finite numbers and infinite numbers, continued to flourish with scholars such as Yativrsabha. He was a contemporary of Varahamihira and of the slightly older Aryabhata. We should also note that the two schools at Kusumapura and Ujjain were involved in the continuing developments of the numerals and of place-valued number systems. The next figure of major importance at the Ujjain school was Brahmagupta near the beginning of the seventh century AD and he would make one of the most major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. It is a sobering thought that eight hundred years later European mathematics would be struggling to cope without the use of negative numbers and of zero.

These were certainly not Brahmagupta's only contributions to mathematics. Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables.

The way that the contributions of these mathematicians were prompted by a study of methods in spherical astronomy is described in [25]:-

The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus. But, by means of suitable constructions within the armillary sphere, they were able to reduce many of their problems to comparison of similar right-angled plane triangles. In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations. ... Of the methods taught by Aryabhata and demonstrated by his scholiast Bhaskara I, some are based on comparison of similar right-angled plane triangles, and others are derived from inference. Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.

Before continuing to describe the developments through the classical period we should explain the mechanisms which allowed mathematics to flourish in India during these centuries. The educational system in India at this time did not allow talented people with ability to receive training in mathematics or astronomy. Rather the whole educational system was family based. There were a number of families who carried the traditions of astrology, astronomy and mathematics forward by educating each new generation of the family in the skills which had been developed. We should also note that astronomy and mathematics developed on their own, separate for the development of other areas of knowledge.

Now a "mathematical family" would have a library which contained the writing of the previous generations. These writings would most likely be commentaries on earlier works such as the Aryabhatiya of Aryabhata. Many of the commentaries would be commentaries on commentaries on commentaries etc. Mathematicians often wrote commentaries on their own work. They would not be aiming to provide texts to be used in educating people outside the family, nor would they be looking for innovative ideas in astronomy. Again religion was the key, for astronomy was considered to be of divine origin and each family would remain faithful to the revelations of the subject as presented by their gods. To seek fundamental changes would be unthinkable for in asking others to accept such changes would be essentially asking them to change religious belief. Nor do these men appear to have made astronomical observations in any systematic way. Some of the texts do claim that the computed data presented in them is in better agreement with observation than that of their predecessors but, despite this, there does not seem to have been a major observational programme set up. Paramesvara in the late fourteenth century appears to be one of the first Indian mathematicians to make systematic observations over many years.

Mathematics however was in a different position. It was only a tool used for making astronomical calculations. If one could produce innovative mathematical ideas then one could exhibit the truths of astronomy more easily. The mathematics therefore had to lead to the same answers as had been reached before but it was certainly good if it could achieve these more easily or with greater clarity. This meant that despite mathematics only being used as a computational tool for astronomy, the brilliant Indian scholars were encouraged by their culture to put their genius into advances in this topic.

A contemporary of Brahmagupta who headed the research centre at Ujjain was Bhaskara I who led the Asmaka school. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara lived the astronomer Lalla, another commentator on Aryabhata.

The ninth century saw mathematical progress with scholars such as Govindasvami, Mahavira, Prthudakasvami, Sankara, and Sridhara. Some of these such as Govindasvami and Sankara were commentators on the text of Bhaskara I while Mahavira was famed for his updating of Brahmagupta's book. This period saw developments in sine tables, solving equations, algebraic notation, quadratics, indeterminate equations, and improvements to the number systems. The agenda was still basically that set by Aryabhata and the topics being developed those in his work.

The main mathematicians of the tenth century in India were Aryabhata II and Vijayanandi, both adding to the understanding of sine tables and trigonometry to support their astronomical calculations. In the eleventh century Sripati and Brahmadeva were major figures but perhaps the most outstanding of all was Bhaskara II in the twelfth century. He worked on algebra, number systems, and astronomy. He wrote beautiful texts illustrated with mathematical problems, some of which we present in his biography, and he provided the best summary of the mathematics and astronomy of the classical period.

Bhaskara II may be considered the high point of Indian mathematics but at one time this was all that was known [26]:-

For a long time Western scholars thought that Indians had not done any original work till the time of Bhaskara II. This is far from the truth. Nor has the growth of Indian mathematics stopped with Bhaskara II. Quite a few results of Indian mathematicians have been rediscovered by Europeans. For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.

Following Bhaskara II there was over 200 years before any other major contributions to mathematics were made on the Indian subcontinent. In fact for a long time it was thought that Bhaskara II represented the end of mathematical developments in the Indian subcontinent until modern times. However in the second half of the fourteenth century Mahendra Suri wrote the first Indian treatise on the astrolabe and Narayana wrote an important commentary on Bhaskara II, making important contributions to algebra and magic squares. The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions. Madhava was from Kerala and his work there inspired a school of followers such as Nilakantha and Jyesthadeva.

Some of the remarkable discoveries of the Kerala mathematicians are described in [26]. These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral. Of particular interest is the approximation to the value of π which was the first to be made using a series. Madhava's result which gave a series for π, translated into the language of modern mathematics, reads

π R = 4R - 4R/3 + 4R/5 - ...

This formula, as well as several others referred to above, were rediscovered by European mathematicians several centuries later. Madhava also gave other formulae for π, one of which leads to the approximation 3.14159265359.

The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. Whish's publication in the Transactions of the Royal Asiatic Society of Great Britain and Ireland was essentially unnoticed by historians of mathematics. Only 100 years later in the 1940s did historians of mathematics look in detail at the works of Kerala's mathematicians and find that the remarkable claims made by Whish were essentially true. See for example [15]. Indeed the Kerala mathematicians had, as Whish wrote:-

... laid the foundation for a complete system of fluxions ...

and these works:-

... abound with fluxional forms and series to be found in no work of foreign countries.

There were other major advances in Kerala at around this time. Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations. These types are all the possible pairs of equations of the following seven forms:

x + y = a, x - y = b, xy = c, x2 + y2 = d, x2 - y2 = e, x3 + y3 = f, and x3 - y3 = g.

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. See [12] for more details.

Now we have presented the latter part of the history of Indian mathematics in an unlikely way. That there would be essentially no progress between the contributions of Bhaskara II and the innovations of Madhava, who was far more innovative than any other Indian mathematician producing a totally new perspective on mathematics, seems unlikely. Much more likely is that we are unaware of the contributions made over this 200 year period which must have provided the foundations on which Madhava built his theories.

Our understanding of the contributions of Indian mathematicians has changed markedly over the last few decades. Much more work needs to be done to further our understanding of the contributions of mathematicians whose work has sadly been lost, or perhaps even worse, been ignored. Indeed work is now being undertaken and we should soon have a better understanding of this important part of the history of mathematics.

"(PIYUSH CONSTANT)

SUM OF EACH DIGIT REMAINS "9"(PIYUSH CONSTANT)
Fri, 2006-07-21 22:34 — piyushdadriwala
SUM OF EACH DIGIT REMAINS SAME(9),NINE
I AM VERY MUCH FOND OF MATHS ,WHATEVER I AM WRITING HERE IS AMAZING,INTERESTING,LEARN IT,VERY SIMPLE.(FOR ANY NO OF DIGITS)
NOW,I HAVE 25 AND 32, MULTIPLE THEM ,NOW YOU CAN MULTIPLE THEM IN FOUR WAYS LIKE THAT(just changing the position)
25*32=800
52*32=1664
25*23=575
52*23=1196
now substract any bigger to any lower you will always get sum of each digit nine.
1664-1196=468(4+6+8=18=1+8=9)
1664-800=864(8+6+4+18=1+8+=9)
1664-575=1089(1+0+8+9=18=1+8=9)
1196-800=396(3+9+6=18=1+8=9)
1196-575=621(6+2+1=9)
800-575=225(2+2+5=9).
this i called "piyush contant"
with lot of regards
piyushdadriwala
www.piyush-g.741.com
pkgdwala@rediffmail.com
in the next topic"what all GODS HAVE COMMON".

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Monday, August 6, 2007

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aristotle

Aristotle (384 BC - March 7, 322 BC) was an ancient Greek philosopher, student of Plato and teacher of Alexander the Great. He wrote many books about physics, poetry, zoology, logic, rhetoric, government, and biology.
Aristotle, along with Plato and Socrates, are generally considered the three most influential ancient Greek philosophers in Western thought. Among them they transformed Presocratic Greek philosophy into the foundations of Western philosophy as we know it. The writings of Plato and Aristotle form the core of Ancient philosophy.
Aristotle placed much more value on knowledge gained from the senses and would correspondingly be better classed among modern empiricists (see materialism and empiricism). He also achieved a "grounding" of dialectic in the Topics by allowing interlocutors to begin from commonly held beliefs (Endoxa); his goal being non-contradiction rather than Truth. He set the stage for what would eventually develop into the scientific method centuries later. Although he wrote dialogues early in his career, no more than fragments of these have survived.
The works of Aristotle that still exist today are in treatise form and were, for the most part, unpublished texts. These were probably lecture notes or texts used by his students, and were almost certainly revised repeatedly over the course of years. As a result, these works tend to be eclectic, dense and difficult to read.
Among the most important ones are Physics, Metaphysics, Nicomachean Ethics, Politics, De Anima (On the Soul) and Poetics.
Their works, although connected in many fundamental ways, are very different in both style and substance.
Aristotle is known for being one of the few figures in history who studied almost every subject possible at the time. In science, Aristotle studied anatomy, astronomy, embryology, geography, geology, meteorology, physics, and zoology.
In philosophy, Aristotle wrote on aesthetics, economics, ethics, government, metaphysics, politics, psychology, rhetoric and theology. He also dealt with education, foreign customs, literature and poetry. His combined works practically comprise an encyclopedia of Greek knowledge.
Early life and studies at the Academy
Aristotle was born at Stageira, a colony of Andros on the Macedonian peninsula of Chalcidice in 384 BC. His father, Nicomachus, was court physician to King Amyntas III of Macedon. It is believed that Aristotle's ancestors held this position under various kings of Macedonia. As such, Aristotle's early education would probably have consisted of instruction in medicine and biology from his father.
About his mother, Phaestis, little is known. It is known that she died early in Aristotle's life. When Nicomachus also died, in Aristotle's tenth year, he was left an orphan and placed under the guardianship of his uncle, Proxenus of Atarneus. He taught Aristotle Greek, rhetoric, and poetry (O'Connor et al., 2004). Aristotle was probably influenced by his father's medical knowledge; when he went to Athens at the age of 18, he was likely already trained in the investigation of natural phenomena.
From the age of 18 to 37 Aristotle remained in Athens as a pupil of Plato and distinguished himself at the Academy. The relations between Plato and Aristotle have formed the subject of various legends, many of which depict Aristotle unfavourably. No doubt there were divergences of opinion between Plato, who took his stand on sublime, idealistic principles, and Aristotle, who even at that time showed a preference for the investigation of the facts and laws of the physical world. It is also probable that Plato suggested that Aristotle needed restraining rather than encouragement, but not that there was an open breach of friendship.
In fact, Aristotle's conduct after the death of Plato, his continued association with Xenocrates and other Platonists, and his allusions in his writings to Plato's doctrines prove that while there were conflicts of opinion between Plato and Aristotle, there was no lack of cordial appreciation or mutual forbearance. Besides this, the legends that reflect Aristotle unfavourably are traceable to the Epicureans, who were known as slanderers. If such legends were circulated widely by patristic writers such as Justin Martyr and Gregory Nazianzen, the reason lies in the exaggerated esteem Aristotle was held in by the early Christian heretics, not in any well-grounded historical tradition.
Aristotle as philosopher and tutor
After the death of Plato (347 BC), Aristotle was considered as the next head of the Academy, a post that was eventually awarded to Plato's nephew. Aristotle then went with Xenocrates to the court of Hermias, ruler of Atarneus in Asia Minor, and married his niece and adopted daughter, Pythia.
In 344 BC, Hermias was murdered in a rebellion, and Aristotle went with his family to Mytilene. It is also reported that he stopped on Lesbos and briefly conducted biological research. Then, one or two years later, he was summoned to Pella, the Macedonian capital, by King Philip II of Macedon to become the tutor of Alexander the Great, who was then 13.
Plutarch wrote that Aristotle not only imparted to Alexander a knowledge of ethics and politics, but also of the most profound secrets of philosophy. We have much proof that Alexander profited by contact with the philosopher, and that Aristotle made prudent and beneficial use of his influence over the young prince (although Bertrand Russell disputes this). Due to this influence, Alexander provided Aristotle with ample means for the acquisition of books and the pursuit of his scientific investigation.
It is possible that Aristotle also participated in the education of Alexander's boyhood friends, which may have included for example Hephaestion and Harpalus. Aristotle maintained a long correspondence with Hephaestion, eventually collected into a book, unfortunately now lost.According to sources such as Plutarch and Diogenes, Philip had Aristotle's hometown of Stageira burned during the 340s BC, and Aristotle successfully requested that Alexander rebuild it. During his tutorship of Alexander, Aristotle was reportedly considered a second time for leadership of the Academy; his companion Xenocrates was selected instead.
Founder and Master of the Lyceum
In about 335 BC, Alexander departed for his Asiatic campaign, and Aristotle, who had served as an informal adviser (more or less) since Alexander ascended the Macedonian throne, returned to Athens and opened his own school of philosophy. He may, as Aulus Gellius says, have conducted a school of rhetoric during his former residence in Athens; but now, following Plato's example, he gave regular instruction in philosophy in a gymnasium dedicated to Apollo Lyceios, from which his school has come to be known as the Lyceum. (It was also called the Peripatetic School because Aristotle preferred to discuss problems of philosophy with his pupils while walking up and down -- peripateo -- the shaded walks -- peripatoi -- around the gymnasium).
During the thirteen years (335 BC­322 BC) which he spent as teacher of the Lyceum, Aristotle composed most of his writings. Imitating Plato, he wrote Dialogues in which his doctrines were expounded in somewhat popular language.
He also composed the several treatises (which will be mentioned below) on physics, metaphysics, and so forth, in which the exposition is more didactic and the language more technical than in the Dialogues. When reported or imitated in writing, "dialogue" labels a form of literature invented by the Greeks for purposes of rhetorical entertainment and instruction, and scarcely modified since the days of its invention.
These writings show to what good use he put the resources Alexander had provided for him. They show particularly how he succeeded in bringing together the works of his predecessors in Greek philosophy, and how he pursued, either personally or through others, his investigations in the realm of natural phenomena.
Pliny claimed that Alexander placed under Aristotle's orders all the hunters, fishermen, and fowlers of the royal kingdom and all the overseers of the royal forests, lakes, ponds and cattle-ranges, and Aristotle's works on zoology make this statement more believable. Aristotle was fully informed about the doctrines of his predecessors, and Strabo asserted that he was the first to accumulate a great library.
During the last years of Aristotle's life the relations between him and Alexander became very strained, owing to the disgrace and punishment of Callisthenes, whom Aristotle had recommended to Alexander. Nevertheless, Aristotle continued to be regarded at Athens as a friend of Alexander and a representative of Macedonia. Consequently, when Alexander's death became known in Athens, and the outbreak occurred which led to the Lamian war, Aristotle shared in the general unpopularity of the Macedonians.
The charge of impiety, which had been brought against Anaxagoras and Socrates, was now, with even less reason, brought against Aristotle. He left the city, saying (according to many ancient authorities) that he would not give the Athenians a chance to sin a third time against philosophy. He took up residence at his country house at Chalcis, in Euboea, and there he died the following year, 322 BC.
His death was due to a disease, reportedly 'of the stomach', from which he had long suffered. The story that his death was due to hemlock poisoning, as well as the legend that he threw himself into the sea "because he could not explain the tides," is without historical foundation.
Very little is known about Aristotle's personal appearance except from hostile sources. The statues and busts of Aristotle, possibly from the first years of the Peripatetic School, represent him as sharp and keen of countenance, and somewhat below the average height. His character - as revealed by his writings, his will (which is undoubtedly genuine), fragments of his letters and the allusions of his unprejudiced contemporaries - was that of a high-minded, kind-hearted man, devoted to his family and his friends, kind to his slaves, fair to his enemies and rivals, grateful towards his benefactors.
When Platonism ceased to dominate the world of Christian speculation, and the works of Aristotle began to be studied without fear and prejudice, the personality of Aristotle appeared to the Christian writers of the 13th century, as it had to the unprejudiced pagan writers of his own day, as calm, majestic, untroubled by passion, and undimmed by any great moral defects, "the master of those who know".
Aristotle's legacy also had a profound influence on Islamic thought and philosophy during the middle ages. The likes of Avicenna, Farabi, and Yaqub ibn Ishaq al-Kind were a few of the major proponents of the Aristotelian school of thought during the Golden Age of Islam.
Methodology
Aristotle defines philosophy in terms of essence, saying that philosophy is "the science of the universal essence of that which is actual".
Plato had defined it as the "science of the idea", meaning by idea what we should call the unconditional basis of phenomena. Both pupil and master regard philosophy as concerned with the universal; Aristotle, however, finds the universal in particular things, and called it the essence of things, while Plato finds that the universal exists apart from particular things, and is related to them as their prototype or exemplar.
For Aristotle, therefore, philosophic method implies the ascent from the study of particular phenomena to the knowledge of essences, while for Plato philosophic method means the descent from a knowledge of universal ideas to a contemplation of particular imitations of those ideas. In a certain sense, Aristotle's method is both inductive and deductive, while Plato's is essentially deductive.
In Aristotle's terminology, the term natural philosophy corresponds to the phenomena of the natural world, which include: motion, light, and the laws of physics. Many centuries later these subjects would later become the basis of modern science, as studied through the scientific method. The term philosophy is distinct from metaphysics, which is what moderns term philosophy.
In the larger sense of the word, he makes philosophy coextensive with reasoning, which he also called "science". Note, however, that his use of the term science carries a different meaning than that which is covered by the scientific method.
"All science (dianoia) is either practical, poetical or theoretical." By practical science he understands ethics and politics; by poetical, he means the study of poetry and the other fine arts; while by theoretical philosophy he means physics, mathematics, and metaphysics.
The last, philosophy in the stricter sense, he defines as "the knowledge of immaterial being," and calls it "first philosophy", "the theologic science" or of "being in the highest degree of abstraction." If logic, or, as Aristotle calls it, Analytic, be regarded as a study preliminary to philosophy, we have as divisions of Aristotelian philosophy (1) Logic; (2) Theoretical Philosophy, including Metaphysics, Physics, Mathematics, (3) Practical Philosophy; and (4) Poetical Philosophy.
Aristotle's Theory of Universals
Aristotle's theory of universals is one of the classic solutions to the problem of universals. Aristotle thought - to put it in a not-very-enlightening way - that universals are simply types, properties, or relations that are common to their various instances.
In Aristotle's view, universals exist only where they are instantiated; they exist only in things (he said they exist in re, which means simply "in things"), never apart from things. Beyond this Aristotle said that a universal is something identical in each of its instances. So all red things are similar in that there is the same universal, redness, in each red thing.
There is no Platonic form of redness, standing apart from all red things; instead, in each red thing there is the same universal, redness.
To further flesh out Aristotle's theory of universals, it is useful to consider how the theory might satisfy the constraints on theories of universals listed in the problem of universals article.
First of all, on Aristotle's view, universals can be multiply instantiated. Aristotle stresses, after all, the one and the same universal, applehood (say), that appears in each apple. Common sense might detect a problem here. (The problem can arise for other forms of realism about universals, however.) Namely, how can we make sense of exactly the same thing being in all of these different objects? That after all is what the theory says; to say that different deserts, the Sahara, the Atacama, and the Gobi are all dry places, is just to say that the exact same being, the universal dryness, occurs at each place. Universals must be awfully strange entities if exactly the same universal can exist in many places and times at once, or so one might think. But maybe that's not so troubling; it seems troubling if we expect universals to be like physical objects, but remember, we are talking about a totally different category of being. So a common defense of realism (and hence of Aristotle's realism) is that we should not expect universals to behave as ordinary physical objects do. Maybe then it is not so strange, then, to say that the exact same universal, dryness, occurs all over the earth at once; after all, there is nothing strange about saying that different deserts can be dry at the same time.
Are Aristotelian universals abstract? And are they, then, what we conceive of when we conceive of abstract objects such as redness? Perhaps. It will help to explain something about how we form concepts, according to Aristotle. We might think of a little girl just forming the concept of human beings. How does she do it? When we form the concept of a universal on Aristotle's theory, we abstract from a lot of the instances we come across. We as it were mentally extract from each thing the quality that they all have in common. So how does the little girl get the concept of a human being? She learns to ignore the details, tall and short, black and white, long hair and short hair, male and female, etc.; and she pays attention to the thing that they all have in common, namely, humanity. On Aristotle's view, the universal humanity is the same in all humans (i.e., all humans have that exact same type in common); and this allows us to form a concept of humanity that applies to all humans.
Are Aristotelian universals the sorts of things we refer to when we use general terms, like 'redness' and 'humanity'? Again, perhaps. The idea is that when we refer to humanity, we refer to the type, human being, that appears identically in each human. We do not refer simply to all the humans, but instead the type, human being, which is the same in each human.
Aristotle's Epistemology
Logic
The Organon is the name given by Aristotle's followers, the Peripatetics, for the standard collection of six of his works on logic. The system of logic described in two of these works, namely On Interpretation and the Prior Analytics, often called Aristotelian logic, is discussed in the article on term logic.
Continued
Non-Aristotelian logic
History
Aristotle "says that 'on the subject of reasoning' he 'had nothing else on an earlier date to speak about'".
However, Plato reports that syntax was thought of before him, by Prodikos of Keos, who was concerned by the right use of words. Logic seems to have emerged from dialectics, the earlier philosophers used concepts like reductio ad absurdum as a rule when discussing, but never understood its logical implications.
Even Plato had difficulties with logic. Although he had the idea of constructing a system for deduction, he was never able to construct one. Instead, he relied on his dialectic, which was a confusion between different sciences and methods. Plato thought that deduction would simply follow from premises, so he focused on having good premises so that the conclusion would follow. Later on, Plato realiszed that a method for obtaining the conclusion would be beneficial. Plato never obtained such a method, but his best attempt was published in his book Sophist, where he introduced his division method.
Analytics and the Organon
What we call today Aristotelian logic, Aristotle himself would have labelled analytics. The term logic he reserved to mean dialectics. Most of Aristotle's work is probably not authentic, since it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into six books at about the time of Christ:
1. Categories 2. On Interpretation 3. Prior Analytics 4. Posterior Analytics 5. Topics 6. On Sophistical Refutations
The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. There is one volume of Aristotle's concerning logic not found in the Organon, namely the fourth book of Metaphysics.
Modal logic
Aristotle is also the creator of syllogisms with modalities (modal logic). The word modal refers to the word 'modes', explaining the fact that modal logic deals with the modes of truth. Aristotle introduced the qualification of 'necessary' and 'possible' premises. He constructed a logic which helped in the evaluation of truth but which was very difficult to interpret.
Science
Aristotelian discussions about science had only been qualitative, not quantitative. By the modern definition of the term, Aristotelian philosophy was not science, as this worldview did not attempt to probe how the world actually worked through experiment. For example, in his book The History of Animals he claimed that human males have more teeth than females. Had he only made some observations, he would have discovered that this claim is false.
Rather, based on what one's senses told one, Aristotelian philosophy then depended upon the assumption that man's mind could elucidate all the laws of the universe, based on simple observation (without experimentation) through reason alone.
One of the reasons for this was that Aristotle held that physics was about changing objects with a reality of their own, whereas mathematics was about unchanging objects without a reality of their own. In this philosophy, he could not imagine that there was a relationship between them.
In contrast, today's "science" assumes that thinking alone often leads people astray, and therefore one must compare one's ideas to the actual world through experimentation; only then can one see if one's ideas are based in reality. This position is known as empiricism or the scientific method.
Aristotle's Metaphysics
Aristotle's four causes: Aristotle names four "causes" of things, but the word cause is not used in the modern sense of "cause and effect", under which causes are events or states of affairs. Rather, the four causes are like different ways of explaining something:
The Material Cause - (That from which it comes) -
This is the material that makes up an object, for example, "the bronze and silver ... are causes of the statue and the bowl." The Material Cause, that out of which the statue is made, is the marble or bronze. The material cause implies the capacity of existence to reside in the substance of the material of which the universe is made. Most scientific inquiry involves that concept, thus ignores the formal cause, touches on the efficient cause, and denies the final cause. Some scientific discussion does imply a teleological cause in biology by suggesting that organisms seek to propagate themselves as a condition of their genome. Survival of the fittest implies a teleological cause or desire to improve future generations.
The Formal Cause (That which it is)
This is the blueprint or the idea commonly held of what an object should be. Aristotle says, "The form is the account (and the genera of the account) of the essence (for instance, the cause of an octave is the ratio two to one, and in general number), and the parts that are in the account."
The Efficient Cause (That which moves it)
This is the person who makes an object, or "unmoved movers" (gods) who move nature. For example, "a father is a cause of his child; and in general the producer is a cause of the product and the initiator of the change is a cause." This is closest to the modern definition of "cause".
The Final Cause (That of which its purpose is)
The final cause is that for the sake of which a thing exists or is done, including both purposeful and instrumental actions and activities. The final cause or telos is the purpose or end that something is supposed to serve, or it is that from which and that to which the change is. This also covers modern ideas of mental causation involving such psychological causes as volition, need, motivation, or motives, rational, irrational, ethical, all that gives purpose to behavior. The final cause of the artist might be the statue itself. (teleology)
Additionally, things can be causes of one another, causing each other reciprocally, as hard work causes fitness and vice versa, although not in the same way or function, the one is as the beginning of change, the other as the goal. [Thus Aristotle first suggested a reciprocal or circular causality as a relation of mutual dependence or action or influence of cause and effect.] Also, Aristotle indicated that the same thing can be the cause of contrary effects, its presence and absence may result in different outcomes.
Aristotle marked two modes of causation: proper (prior) causation and accidental (chance) causation. All causes, proper and incidental, can be spoken as potential or as actual, particular or generic. The same language refers to the effects of causes, so that generic effects assigned to generic causes, particular effects to particular causes, operating causes to actual effects. Essentially, causality does not suggest a temporal relation between the cause and the effect.
All further investigations of causality will consist of imposing the favorite hierarchies on the order causes, such as final > efficient> material > formal (Thomas Aquinas), or of restricting all causality to the material and efficient causes or to the efficient causality (deterministic or chance) or just to regular sequences and correlations of natural phenomena (the natural sciences describing how things happen instead of explaining the whys and wherefores).
Modes of Causation
Aristotle states two modes of causation:
Proper Causation: Things take place for the sake of something, and the result is that which is intended.
Accidental Causation: Things that take place not out of necessity, i.e. things that take place by chance/coincidence. This cause is indeterminable.
Chance lies in the realm of accidental causes. It is "from what is spontaneous" (but note that what is spontaneous does not come from chance). For a better understanding of Aristotle's conception of "chance" it might be better to think of "coincidence": Something takes place by chance if a person sets out with the intent of having one thing take place, but with the result of another thing (not intended) taking place.
For example: A person seeks donations. That person may find another person willing to donate a substantial sum. However, if the person seeking the donations met the person donating, not for the purpose of collecting donations, but for some other purpose, Aristotle would call the collecting of the donation by that particular donator a result of chance. It must be unusual that something happens by chance. In other words, if something happens all or most of the time, we cannot say that it is by chance.
However, chance can only apply to human beings. According to Aristotle, chance must involve choice (and thus deliberation), and only humans are capable of deliberation and choice. "What is not capable of action cannot do anything by chance" (Physics, 2.6).
The Five Elements
Fire which is hot and dry.
Earth which is cold and dry.
Air which is hot and wet.
Water which is cold and wet.
Aether which is the divine substance that makes up the heavens
These four elements interchange (i.e. Fire, Air, Water, Earth), while aether is on its own. The Sun keeps this cycle going. God keeps the Sun going (and thus the Sun is eternal).
Aristotle's Ethics
Although Aristotle wrote several works on ethics, the major one was the Nicomachean Ethics, which is considered one of Aristotle's greatest works; it discusses virtues. The ten books which comprise it are based on notes from his lectures at the Lyceum and were either edited by or dedicated to Aristotle's son, Nicomachus.
Aristotle believed that ethical knowledge is not certain knowledge (like metaphysics and epistemology) but is general knowledge. Also, as it is not a theoretical discipline, he thought a person had to study in order to become "good." Thus, if a person was to become virtuous, they could not simply study what virtue is, they had to actually do virtuous activity. In order to do this, Aristotle had to first establish what was virtuous. He began by determining that everything was done with some goal in mind and that goal is 'good.' The ultimate goal he called the Highest Good.
Aristotle contested that happiness could not be found only in pleasure or only in fame and honor. He finally finds happiness "by ascertaining the specific function of man. But what is this function that will bring happiness?
To determine this, Aristotle analyzed the soul and found it to have three parts: the Nutritive Soul (plants, animals and humans), the Perceptive Soul (animals and humans) and the Rational Soul (humans only). Thus, a human's function is to do what makes it human, to be good at what sets it apart from everything else: the ability to reason or Nous. A person that does this is the happiest because they are fulfulling their purpose or nature as found in the rational soul. Depending on how well they did this, Aristotle said people belonged to one of four categories: the Virtuous, the Continent, the Incontinent and the Vicious.
Aristotle believes that every ethical virtue is an intermediate condition between excess and deficiency. This does not mean Aristotle believed in moral relativism, however. He set certain emotions (e.g., hate, envy, jealousy, spite, etc.) and certain actions (e.g., adultery, theft, murder, etc.) as being always wrong, regardless of the situation or the circumstances.
Aristotelian Ethics
Nicomachean Ethics
In Nicomachean Ethics, Aristotle focuses on the importance of continually behaving virtuously and developing virtue rather than committing specific good actions. This can be opposed to Kantian ethics, in which the primary focus is on individual action. Nicomachean Ethics emphasizes the importance of context to ethical behavior - what might be right in one situation might be wrong in another. Aristotle believed that happiness is the end of life and that as long as a person is striving for goodness, good deeds will result from that struggle, making the person virtuous and therefore happy.
Nicomachean Ethics
Aristotle's Critics
Aristotle has been criticised on several grounds.
His analysis of procreation is frequently criticised on the grounds that it presupposes an active, ensouling masculine element bringing life to an inert, passive, lumpen female element; it is on these grounds that some feminist critics refer to Aristotle as a misogynist.
At times, the objections that Aristotle raises against the arguments of his own teacher, Plato, appear to rely on faulty interpretations of those arguments.
Although Aristotle advised, against Plato, that knowledge of the world could only be obtained through experience, he frequently failed to take his own advice. Aristotle conducted projects of careful empirical investigation, but often drifted into abstract logical reasoning, with the result that his work was littered with conclusions that were not supported by empirical evidence; for example, his assertion that objects of different mass fall at different speeds under gravity, which was later refuted by John Philoponus. Credit is often given to Galileo, even though Philopinus lived centuries before him.
In the Middle Ages, roughly from the 12th century to the 15th century, the philosophy of Aristotle became firmly established dogma. Although Aristotle himself was far from dogmatic in his approach to philosophical inquiry, two aspects of his philosophy might have assisted its transformation into dogma. His works were wide-ranging and systematic so that they could give the impression that no significant matter had been left unsettled. He was also much less inclined to employ the sceptical methods of his predecessors, Socrates and Plato.
Some academics have suggested that Aristotle was unaware of much of the current science of his own time, and that he was a far lesser mathematician than many of his learned contemporaries.
Aristotle was called not a great philosopher, but "The Philosopher" by Scholastic thinkers. These thinkers blended Aristotelian philosophy with Christianity, bringing the thought of Ancient Greece into the Middle Ages. It required a repudiation of some Aristotelian principles for the sciences and the arts to free themselves for the discovery of modern scientific laws and empirical methods.
The Western mind is "Aristotelian". By this we mean that it formats the external world into factual and "scien"-tific categories. (By "Scien"-tific we mean that something is knowable or known. Latin scientia = knowledge).
Under the premise of external categorization, the Aristotelian mind has come to equate "experience" with the unified chronical and spatial ontological structure that is the "external" universe -- visible, audible and sensible by the handful of our common, well-identified senses.
By so equating the two, the Aristotelian mind is fully confident, or fully "positive" of the meanings of its utterances and the purposes of all actions. That is to say, it dismisses the possibility of dubious meanings as interpreted by subjects that are at variance in perspectives or phenomenology, and it dismisses the importance of anything other than an objectively defined "purpose" to an action.
Therefore, the Aristotelian mind assumes that when subject A utters "I am X," he or she is referring to the same experience and is expressing the same purpose as subject B who also utters "I am X."
Major Works
The extant works of Aristotle are broken down according to the five categories in the Corpus Aristotelicum.
The Corpus Aristotelicum refers to the traditional ordering and categorization of the works of Aristotle, dating back to the 2nd century. Although the works were all considered to be genuine until recently, modern scholarship has cast doubts on the authenticity on many of the texts. The only major work of Aristotle's not in the Corpus Aristotelicum is the Constitution of the Athenians.
Not all of these works are considered genuine, but differ with respect to their connection to Aristotle, his associates and his views. Some, such as the Athenaion Politeia or the fragments of other politeia are regarded by most scholars as products of Aristotle's "school" and compiled under his direction or supervision. Other works, such On Colours may have been products of Aristotle's successors at the Lyceum, e.g., Theophrastus and Straton. Still others acquired Aristotle's name through similarities in doctrine or content, such as the De Plantis, possibly by Nicolaus of Damascus. A final category, omitted here, includes medieval palmistries, astrological and magical texts whose connection to Aristotle is purely fanciful and self-promotional. Those that are seriously disputed are marked with an asterisk.