17.
criticism of the Renaissance: *
Continuity rather than revolution is more apparent between 1400-1600. Throughout the Middle Ages scholars looked back at the Greeks & Romans as the highest possible authorities (other than the Bible). The 12th-century rediscovery of ancient texts lost (until found in Arabic translations) stimulated this admiration. However where Medieval scholars saw themselves as inferior to the Ancients, Renaissance humanists saw themselves as their equals. This attitude led them to denigrate their immediate predecessors. Petrarch despised medieval writers & instead copied Greek & Roman writers. Renaissance architects rejected Gothic styles & copied Roman styles. Even the terms “Medieval” & “Middle Ages” were invented by Renaissance writers to belittle the interruption (Medieval ages) between the great Classical Age and their own supposed revival. In fact the Renaissance was a continuation & outgrowth of the Middle Ages; the revival of ancient learning took place mainly in the Middle Ages & the economic progress making possible Renaissance art & architecture came from Medieval expansion & innovation. Continuity is stronger than radical change. Spengler would add to the above argument his belief that the Faustian Culture’s ability to truly emulate or understand Classical culture was very limited. Indeed it is one of his repeated themes that different Cultures cannot understand each other. The Renaissance artists owed almost nothing to Greece & Rome, their achievements were the result of their own genius, not simply copies of an older & dead Culture which they did not and could not understand.
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18.
Indians...number: *
Its development is hard to trace. Mathematics was a tool for astronomers with no independent existence. Priests dominated the field, theories written in poetry, proofs not saved & problems transcribed in poetic words as riddles & puzzles (solutions by the astronomer would give great social kudos). Where the Greeks focused on geometry, Hindus focused on arithmetic, numerical symbolism & the science of number & algebra. India & the Greeks did share ideas however.
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The earliest known astronomer (and mathematician) is Aryabhatta (born 476 AD); Hindu development seems to have peaked with Brahmagupta (born 598 AD). The big advance in Hindu number is the concept of position in written numbers though we have no idea who or when it was invented. The 9 figures for units was invented first & then the symbol for zero & the concept of positon later. [Ceylon supports this. In the 3rd century AD Hindu Culture spread to this island; Culture in Ceylon remained static & preserved the early image. We find 9 figures for units, 9 for ten, 1 for 100 and 1 for 1000. With these 20 characters a number up to 9999 can be written. The notation of Aryabhatta is similar to the Ceylon notation]. Words were used in place of numeric notation but the idea of position is evident. Words allowed numbers to be memorized. Mythically the Buddha was able to use & name a number to 15 digits. It made arithmetic operations easier. Aryabhatta speaks of division as the inversion of multiplication. Areas of study included interest, discounts, partnership, allegation, summation of arithmetical & geometric series, determination of combinations & magic squares as well as chess. They used the “rule of 3” and the “false position” concept.
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Vedic number developed symbols for operations: addition juxtaposed the words; subtraction used a dot over the subtrahend, multiplication indicated with the word “product”; division placed the divisor beneath the dividend; square root used the word “ka” (for irrational); an unknown was designated by the word “unknown quantity”, for several unknowns each had a distinct name using colours (black, blue, yellow, red, green) employing the initial syllable of each colour.
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They recognized negative & positive numbers (from the idea of possession & debt) & knew movement along a line (+ and –). They observed that a quadratic always had 2 roots & knew but did not use or admit negative roots. Unlike the Greek limitation to magnitude (meaning no negative numbers) Hindu numbers freely moved from magnitude to number & number to magnitude. And unlike the Greeks they were not limited to rational numbers but used both rational & irrational. Their algebra consisted of arithmetic operations applied to complex magnitudes, both rational & irrational numbers and for space magnitudes. Some Greek influence is observed in their generalized & improved solutions for linear & quadratic equations.
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They made great progress in their treatment of indeterminate equations. Here the Hindu mind was at home and developed general solutions to these types of problems. Diophantus (whom Spengler would regards as Magian) analysed very specific indeterminate equation problems & came up with a specific solution, he stopped at a single rational answer. The Hindus looked to find all possible solutions. Aryabhatta gives solutions to linear equations (of the form ax ± by = c ; abc integers) using the pulveriser rule (no proof is given). These equations grew out of problems in astronomy, such as determining the time when a certain constellation of the planets occur. With indeterminate quadratic equations, for the solution xy=ax + by +c they used decomposing. Even more advanced was their solution for the quadratic equation cy^3=ax^3 + b. They recognized y^2 = ax^2+1 as the fundamental problem for indeterminate quadratics and solved it using the cyclic methods. Their rule allowed for an indefinite number of solutions. These methods & solutions were applied to astronomy. This use of the cyclic method was a massive advance,far beyond the Greeks or even the Arabs.
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19.
Brahmanic soul: *
Brahma is often identified with the Vedic god Prajapati, also to Kama and Hiranyagarbha (the cosmic egg). He is prominent in the post-Vedic Hindu epics (often conflated with Puranas) & the mythologies in the Puranas (ancient Hindu texts eulogizing various deities, primarily the divine Trimurti). One of the trinity of supreme divinity in Hinduism with Vishnu & Shiva, although ancient Hindu scriptures also mention multiple other trinities of gods or goddesses which do not include Brahma. In the Puranas he is described as emerging from a lotus, connected to the navel of Lord Vishnu; these texts also suggest he is born from Shiva. With other deities he is sometimes viewed as a form (saguna) of the otherwise formless (nirguna) Brahman, the ultimate metaphysical reality in Vedantic Hinduism.
see illustration.
This illustration of Brahma is Pahari art, about 1700 A.D, India.
