Gods, eyes, elephants, serpents, fires, three, qualities, Vedas, nakshatras, elephants, arms: the wise have said that this is the measure of the circumference when the diameter of the circle is nine nikharvas.There are
So we can compute:
which is good to 11 decimal places.
Reference:
1. H.S.White, "Review of Kim Plofker, Mathematics in India", The Mathematical Intelligencer, Volume 32, Number 2, 2010.
That's the problem with working in the foreign exchange markets. Everything one sees is coloured through an exchange rate prism. Fortunately, a quick perusal of the newspaper content disabused me of any foreign exchange related notions. 'Lair' was a typo. That should have read 'Rupiah is a liar.' And, of course, Rupiah is the first name of Mr Sata's opponent at the Mpulungu parliamentary by-election.
Mr Sata added, "I am younger to Rupiah Banda by only six months but I am honest."
He came to talk to us about Climate Change, the facts and the interpretations, the factions and the policking. A good time was had by all.
Here's a quick summary. The good professor pointed out that
- the facts about global warming are not in dispute, but only (mis)interpretations by various vested interests.
- even those who supposedly agree with global warming, e.g. Greens, do not necessarily support the hard choices required to lessen the impact of greenhouse gases (e.g. nuclear power).
- both parties - Deniers and Believers - often have identical rhetoric as regards solutions, but will just not listen to each other because of fundamental biases.
- energy costs are roughly 5% of world GDP because gas/petroleum is so cheap; any replacement available now will require maybe 20% of world GDP to provide the same amount of energy, so unless radically cheaper methodologies appear, there's little possibility to switch out of polluting fuels.
- to get public attention to the problem, it's not enough for scientists to talk loudly and slowly as if to slow foreigners, but to get public figures involved in communicating the issues.
- people need to think outside of their own immediate experience to realise that 'average rising temperatures' doesn't mean that temperatures will be rising everywhere: last winter was colder than the average in Western Europe and the eastern seaboard of the US, but at the same time, most of the rest of the planet was having much warmer times.
All in all, an intellectually enlightening and emotionally depressing presentation.
There were few people in other words who, of their own accord, preferred to be vegetarians.
St. Francis of Assisi was likely one of them (set caged birds free and all that). Another was Leonardo da Vinci.
By the time Leonardo was in his fifties, he had spent decades eating only fruit, nuts and vegetables. His exposure to the horrors of war rendered him extremely sympathetic to any sort of pain, and he considered it a sin to harm any living creatures. Unlike others of strong conviction, though, he didn't force the view on his followers, and he paid out enough of his hard-earned cash to keep his dependants well-fed with meat.
But his vegetarianism - as his genius - became widely known. Praise came - would you believe it - from as far away as India, from where a Florentine traveller named Andrea Corsali wrote:
a gentle people called Guzzarati who do not feed on anything that has blood, nor will they allow anyone to hurt a living thing, like our Leonardo da Vinci.Reference
Peter Strathern, The Artist, The Philosopher and The Warrior
It appears that a man called Albert of Saxony wrote a little treatise in 1350 consisting of various proofs and assertions on the possibility of squaring the circle. His book was called Quaestio de quadratura circuli (Question on the Quadrature of the Circle), and he made some pithy observations such as the following:
If there could not be given a square equal to a circle, it would follow that there would take place passage from "greater" to "lesser," or from extreme to extreme, through all the means without ever arriving at "equal" or "middle." But this is false. Therefore, I prove the consequence. For let there be one square inscribed in a circle and let this square begin to be continually and uniformly increased until it becomes larger than the circle. If, therefore, it was at some time equal to the circle, we have the proposition; if not, then passage has been made from "lesser" to "greater" with respect to that circle without ever arriving at "equal."1
Now I won't claim that this is absolutely rigorous, although it's fairly clear that if you assume that a square can be stretched continuously, then a square inscribed in a circle will at some point exceed the size of circle, and a fundamental consequence of continuity is that it will do so without any sudden breaks. So, in fact, there is a square with exactly the same area as a given circle.
We know this is true. If the area of a circle with radius r is pr2, then the side of the square with the same area will be r√p. Every real number has a square root, so we are good here.
The question that Albert was hoping to answer - I suspect - was the long-standing one of if it's possible to construct the equivalent square using only a straightedge and a compass. This was one of the classical problems of mathematics, known as far as back as the Greeks (Archimedes had provided an incorrect solution), and possibly even earlier.
A century or so after Albert, Leonardo da Vinci put his fecund imagination to the problem. He had constructed very clever mechanical means for squaring the circle, but as his friend (and math teacher) Luca Pacioli pointed out, these were mere approximations, and not true constructions. In fact, they were not even original. Stung by this criticism, Leonardo (sometime in 1503) decided to solve the problem once and for all. From his notebooks, it is evident that he had spent time before this on the issue, trying out one mechanical method after another. That night in November, however, he resolved not to get up from his desk until he had settled the question.
It is possible to trace Leonard's series of ingenious and beautiful designs, as he tried to improve upon Archimedes' faulty solution - until he finally cracked it! In the margin he records the exact time of his discovery:
“On the night of St Andrew's Day I eventually finished squaring the circle: by then my candles were finished, the night was finished, and so was the paper I was writing on. This conclusion came to me at the end of the final hour of the night.”
Alas, he was deluded.2
The reasons for his failure (and the failure of every other mathematician or charlatan who attempted it (and continue to do so to this day)) were to become apparent only 400 years later. In 1892, Lindemann proved that p is a transcendental number. In other words, there is no algebraic equation that has p as its root. Every straightedge-and-compass construction can be translated into an algebraic equation. Therefore, there is no straightedge-and-compass construction that squares a circle.
References
1. Mathematical Intelligencer, Volume 1, Number 3, 1978/79.
2. Paul Strathern, The Artist, The Philosopher and The Warrior, Vintage Books, 2010, London.
3. Tom Pastorello, Leonardo Squared The Circle! – Da Vinci’s Secret Solution in the Vitruvian Man Decoded,
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