Wednesday, November 24, 2010

Essay on Geometry

Essay on Geometry

Jean-Victor Poncelet fought for Napoleon’s France against Russia in the battle at Krosnoy. Unfortunately for him, he was taken prisoner. He survived the subsequent tortuous years through luck alone. During his rehabilitation, and still a prisoner of war, he eagerly resumed his study of Mathematics, at the hospital at Saratov. He had been a pupil of and interested in the work of, Gaspard Monge. However, being isolated in Saratov, he was uninformed about the work that Monge and peers were publishing. He set out to document all he knew about mathematics in Saratov. Detailed notebooks circulated amongst fellow prisoners keen at continuing their studies.

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Poncelet was surprised to find that he remembered, from before his academic hiatus, all but the elementary parts of calculus and algebra. They had remained most intensely in his mind. Also, that he could construct advanced results in mechanics and other topics from his basic understanding of purely geometrical theorems. All that had happened made him determined to document what he knew. This helped to direct his further learning and exploration. He believed in the eternal truths of geometry. He had found fundamental theorems that inspired him to continue to define the general principles of central projection of figures and conic sections. The consequences of what had happened to him gave him the motivation to work towards a more fundamental use of geometry, hence his rediscovery of projective geometry.

His hopes for the work were to make geometry useful and to inspire the working class and the youth of schools. He wanted to inspire people to come to love the eternal truths of science. His work was intended to be less about detail and information and more about illustrating sources. His hopes were to perfect a method of discovery and proof in elementary geometry. He wished to demonstrate and promote algebraic analysis (analytical geometry), in direct opposition to synthetic geometry.

In order to enlighten people to his new way of thinking he presented people with a non-metrical geometry. He was a critic of synthetic geometry. His beliefs were that in some way the apparent necessity to visualise or imagine objects in synthetic geometry halted intellectual discovery. He saw that most people stopped making any sort of comments on objects once these objects ceased to have absolute or physical existence.

Poncelet’s work at its most rigorous was possibly the more mundane statements he made. For once we get an elegant proof and something genuinely projective – the arguments leading up to the definition of a ‘supplementary conic’.

Poncelet wrote many audacious claims. His development of the pole and polar lines associated with conics led to the principle of duality, the power of duality was unquestionable to all, but what was controversial was the boldness of his claims. A lot of what he said was right, but other bits misguided. Often his ideas were not precise enough, take, for example when dualising a curve (degree higher than 2) – no way of dualising twice would return you to the original curve which was in essence the beauty of duality in other problems.

In 1820, a council consisting of Arago, Poisson and Cauchy, (amazing mathematicians), was set up. Its’ purpose; to report on what Poncelet had discovered and written. They saw the power of what he was doing, but reported that his bold induction was only useful if used with guidance and trusting the methods too much caused problems. Not all of what he said was right but the simplicity and power of his thought process was inspiring. Poncelet took a lot credit for his work on projective geometry and duality, notwithstanding criticisms. I believe the politics of the day was more influential on the kudos his work receives than one would hope for. Ultimately though it seems, projective geometry is testimony to how thinking evolved in civilisation through this era. And Poncelet was at the forefront of this topic and its’ progression. For that he should be commended.

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