Moving forward to about 1450, Cusa attempted to prove that the circle could be squared by a plane construction. Although his method of averaging certain inscribed and circumscribed polygons is quite fallacious, it is one of the first serious attempts in 'modern' Europe to solve the problem. Again it is worth commenting that the ancient Greeks basically knew that the circle could not be squared by plane methods, although they stood no chance of proving it. Regiomontanus, who brought a new impetus to European mathematics, was quick to point out the error in Cusa's arguments. The mechanical methods of the Greeks certainly appealed to Leonardo who thought about mathematics in a very mechanical way. He devised several new mechanical methods to square the circle. Many mathematicians in the sixteenth century studied the problem, including Oronce Fine and Giambattista della Porta. The 'proof' by Fine was shown to be incorrect by Pedro Nunes soon after he produced it. The beginnings of the differential and integral calculus led to an increased interest in squaring the circle, but the new era of mathematics still produced fallacious 'proofs' of plane methods to square the circle. One such false proof, given by Saint-Vincent in a book published in 1647, was based on an early type of integration. The problem was still providing much impetus for mathematical development. James Gregory developed a deep understanding of infinite sequences and convergence. He applied these ideas to the sequences of areas of the inscribed and circumscribed polygons of a circle and tried to use the method to prove that there was no plane construction for squaring the circle. His proof essentially attempted to prove that π was transcendental, that is not the root of a rational polynomial equation. Although he was correct in what he tried to prove, his proof was certainly not correct. However, others such as Huygens, believed that π was algebraic, that is that it is the root of a rational polynomial equation. There was still an interest in obtaining methods to square the circle which were not plane methods. For example Johann Bernoulli gave a method of squaring the circle through the formation of evolvents and this method is described in detail in [12]. The historian of mathematics, Montucla, made squaring the circle the topic of his first historical work published in 1754. This was written at a time long before the problem was finally resolved, so is necessarily very outdated. The work is, however, a classic and still well worth reading. A major step forward in proving that the circle could not be squared using ruler and compasses occurred in 1761 when Lambert proved that π was irrational. This was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass. It only led to a greater flood of amateur solutions to the problem of squaring the circle and in 1775 the Paris Académie des Sciences passed a resolution which meant that no further attempted solutions submitted to them would be examined. A few years later the Royal Society in London also banned consideration of any further 'proofs' of squaring the circle as large numbers of amateur mathematicians tried to achieve fame by presenting the Society with a solution. This decision of the Royal Society was described by De Morgan about 100 years later as the official blow to circle-squarers. The popularity of the problem continued and there are many amusing stories told by De Morgan on this topic in his book Budget of Paradoxes which was edited and published by his wife in 1872, the year after his death. De Morgan suggests that St Vitus be made the patron saint of circle-squarers. This is a reference to St Vitus' dance, a wild leaping dance in which people screamed and shouted and which led to a kind of mass hysteria. De Morgan also suggested the term 'morbus cyclometricus' as being the 'circle squaring disease'. Clearly De Morgan found himself having to try to persuade these circle-squarers that their methods were incorrect, yet many stubbornly held to their views despite the best efforts of the professional mathematicians. For example a certain Mr James Smith wrote several books attempting to prove that � = 25 8 π= 8 25 ​ . Of course Mr Smith was able to deduce from this that the circle could be squared but neither Hamilton, De Morgan nor others could convince him of his errors. The final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is it is not the root of any polynomial equation with rational coefficients. The transcendentality of π finally proves that there is no ruler and compass construction to square the circle. One might imagine that this would be the end of interest in the problem of squaring the circle, but this was certainly not the case. It neither prevented the stream of publications claiming that π had some simple rational value, nor did it prevent the stream of publications of quite correct constructions to approximately square the circle with ruler and compass. As an example of the former type of claim, the New York Tribune published a letter in 1892 in which the author claimed to have rediscovered a secret going back to Nicomedes which proved that π = 3.2. Perhaps more surprising is the fact that there were many who were totally convinced by this letter and firmly believed thereafter that π = 3.2. Among the correct approximate constructions to square the circle was one by Hobson in 1913. This was a fairly accurate construction which was based on constructing the approximate value of 3.14164079... for π instead of 3.14159265.... . More remarkable, however, was the ruler and compass constructions published by Ramanujan. In the Journal of the Indian Mathematical Society in 1913 in a paper named Squaring the circle Ramanujan gave a construction which was equivalent to giving the approximate value of 355 113 113 355 ​ for π, which differs from correct value only in the seventh decimal place. He ended the paper with the following:- Note.- If the area of the circle be 140,000 square miles, then [the side of the square] is greater than the true length by about an inch. Among other constructions given by Ramanujan in 1914 (Approximate geometrical constructions for π, Quarterly Journal of Mathematics XLV (1914), 350-374) was a ruler and compass construction which was equivalent to taking the strange yet remarkable approximate value for π to be ( 9 2 + 1 9 2 / 22 ) 1 / 4 (9 2 +19 2 /22) 1/4 . Now this is 3.1415926525826461253.... which differs from π only in the ninth decimal place (π = 3.1415926535897932385...). For a circle of diameter 8000 miles, the error in the length of the side of the square constructed was only a fraction of an inch. '