He entered the University of Leiden in the autumn of 1668 and there he studied mathematics, philosophy, physics and medicine. He also matriculated in the law faculty in Leiden on 8 June 1669 but seems not to have had much of an interest in that topic. He learnt of the latest advances in medicine such as Harvey's theory concerning the circulation of blood. He also took private lessons from Pieter van Schooten (half-brother of the famous mathematician Frans van Schooten) who introduced him to both the mathematics and philosophy of Descartes. In 1672 war broke out between Holland and France and Tschirnhaus enlisted in the student force. He did not see active service but, nevertheless, it meant that he had an eighteen-month interruption to his studies. Tschirnhaus began a European tour in 1674. Setting out from Kieslingswalde he first returned to Leiden where his school friend Pieter van Gent introduced him to Baruch de Spinoza. Tschirnhaus became a regular participant in the debating club which Spinoza ran and this gave him the desire to run an Academy which he later tried to set up. We note in passing that he became more than a life-long friend to van Gent: he was also his patron to the extent that he later paid him for his services as a correspondent, editor and translator. With a letter of recommendation from Spinoza, Tschirnhaus visited England in May 1675 where he met Oldenburg the secretary of the Royal Society. Oldenburg introduced him to, among others, Robert Boyle, Denis Papin, and Isaac Newton. He also met John Collins in London and John Wallis in Oxford. He showed Collins and Wallis his methods for solving equations, but these turned out to be special cases of known results. With a letter of recommendation from Oldenburg, he went to Paris in the autumn of 1675 where he remained for a while after meeting Leibniz and Huygens. Nadler  quotes a letter from G H Schuller to Spinoza written in the autumn of 1675 which describes Tschirnhaus meeting Leibniz in Paris:-
[Tschirnhaus] has met [in Paris] a man named Leibniz of remarkable learning, most skilled in the various sciences and free from the common theological prejudices. [Tschirnhaus] has established a close friendship with him, based on the fact that like him he is working at the problem of the perfecting of the intellect, and indeed he considers there is nothing better or more important than this. In ethics, he says, Leibniz is most practiced, and speaks solely from the dictates of reason uninfluenced by emotion. He adds that in physics and especially in metaphysical studies of God and the Soul he is most skilled ...Leibniz arranged for Tschirnhaus to study the unpublished papers of Descartes and he also was given access to unpublished papers by Pascal and Roberval. After studying these, Tschirnhaus reported his findings in letters to G H Schuller, Pieter van Gent and Spinoza. He met François Villette and observed numerous experiments he was carrying out on burning mirrors and melting of minerals. While in Paris, Tschirnhaus taught one of Jean-Baptiste Colbert's sons but, as Tschirnhaus did not know French, the lessons had to be in Latin. In November 1676 he left Paris, accompanying Count Nimpsch of Silesia. They first travelled to Lyon where Tschirnhaus again met Villette and carried out experiments with him on burning mirrors. He then travelled to Turin, Milan, Venice, Bologna and Rome. Everywhere he went he made contact with the leading scientists including Athanasius Kircher and Alfonso Borelli in Rome. His travels continued with visits to Naples, Sicily, Milan, and Geneva before returning in 1679 to Paris, The Hague (where he visited Huygens) and Hanover (where he visited Leibniz). While making this long journey, Tschirnhaus continued to report his observations and discoveries to Leibniz by letter, receiving helpful replies. For example, on 30 April 1678 Tschirnhaus wrote a long letter to Leibniz from Rome (see ). In it he discussed several mathematical questions including the solution of higher equations. He proposed that algebra is a wide-ranging subject which contains combinatorics as a part. He wrote:-
Many people quite falsely believe that the art of combinatorics is a separate science, to be mastered before algebra and other sciences. Indeed, some people believe that there is more in the art of combinatorics than in the art commonly called algebra; in other words, that the daughter knows more than the mother. But it is certainly obvious, from the composition of powers alone if by nothing else, that the art of combinatorics is mastered through algebra.Leibniz replied:-
... your words are undoubtedly aimed at me, for the 'many' who, as you say, think in this way are few, I believe, beside myself. However, I believe that your opinion is right because you do not seem to have understood me. For if you hold the art of combinatorics to be the science of finding the number of variations, I freely admit that it is subordinate to the science of numbers and consequently to algebra, since the science of numbers is also subordinate to algebra. ... But for me the art of combinatorics is in fact something far different, namely, the science of forms or of similarity and dissimilarity, while algebra is the science of magnitude or of equality and inequality.In his letter Leibniz also criticises Tschirnhaus's solution of algebraic equations. He also sets out some of the fundamental principles of the calculus that he has developed.
After ending his long journey in 1679, Tschirnhaus lived for a time back in his home town of Kieslingswalde. He worked on the construction of circular and parabolic mirrors, aided by his mechanic Johann Hoffmann. These allowed him to obtain high temperatures by focussing sunlight. He was soon on his travels again, however, going to Paris via Holland and Belgium in 1680. He made a third visit to Paris in 1682, and on 22 July he was elected to the Académie des Sciences. He had hoped for a pension to give him financial freedom to continue his scientific studies, but none was forthcoming. Also in 1682 he married Elisabeth Eleonore von Lest in Kieslingswalde. She was the daughter of an important member of the court of the Elector of Saxony. In 1684 Tschirnhaus's father died leaving him the family estate at Kieslingswalde. Tschirnhaus's wife took over managing the estate to allow him the time to continue his scientific researches.
Tschirnhaus worked on the solution of equations and the study of curves. He discovered a transformation which, when applied to an equation of degree n, gave an equation of degree n with no term in xn-1 and xn-2. We have indicated above that he had already discussed his methods for solving equations with Leibniz who had pointed out difficulties. Nevertheless Tschirnhaus published his transformation in Acta Eruditorum in 1683 and, in this article, showed how it could be used to solve the general cubic equation. However, his belief that the method would allow an equation of any degree to be solved is false as had already been pointed out to him by Leibniz. He also studied catacaustic curves in 1682, these being the envelope of light rays emitted from a point source after reflection from a given curve. His work on curves is remembered since a sinusoidal spiral is named after him. Around this time Tschirnhaus was planning to write a major work explaining his philosophy. In 1682 he sent a scheme of what he intended to Huygens but it was six more years before the work came to fruition. In 1686 he published Medicina corporis, then in the following year he published Medicina mentis. The two were put together as a single volume which also appeared in 1687, then a second edition was published in 1695 under the title Medicina mentis sive artis inveniendi praecepta generali. Van Peursen writes :-
He drew lines already traced by rationalist thinkers such as Descartes, Malebranche, and Spinoza. His originality lies in the effort to correct these philosophies. He did this by placing his whole philosophy in the perspective of invention.Tschirnhaus argues that :-
... a person can perform intellectual and other operations without knowing how they actually work. Tschirnhaus frequently gives the example of the way in which we use our hands without any knowledge of their physiological structure. Thus we can admire the manual ability and skill of a watchmaker who does not know anything at all about the way in which his hands function. The whole approach of Tschirnhaus's philosophy is based on this idea, so that the readers can practice the ars inveniendi in a most natural and even naive way.In 1700 Tschirnhaus published Gründliche Anleitung zu nützlichen Wissenschaften which was praised by Leibniz and greatly influenced Christian Wolff.
Hofmann  writes of Tschirnhaus's deep interest in mathematics:-
Tschirnhaus exhausted his mathematical talents in searching for algorithms. Lacking insight into the more profound relations among mathematical propositions, he was too ready to assert the existence of general relationships on the basis of particular results that he obtained. Further he was unwilling to accept suggestions directly from other mathematicians, although he would later adopt them as his own inventions and publish them as such. This tactic led to bitter controversies with Leibniz, Huygens, La Hire and Jacob Bernoulli and Johann Bernoulli, and it ultimately cost him his scientific reputation.The dispute with Leibniz referred to in this quotation took place in 1682-84 and involved the possibility of the algebraic quadrature of algebraic curve. After the publication of Medicina mentis, Fatio de Duiller correctly claimed that a method presented by Tschirnhaus in the book to find tangents to curves was incorrect. This argument also went on for a couple of years during 1687-89.
As well as his work in philosophy and mathematics, Tschirnhaus was a scientist, and among other things, he experimented making porcelain from clay mixed with fusible rock in the 1680s. Much of his experimenting involved the use of burning mirrors with which he was able to generate higher temperatures than had previously been produced. In 1694 he announced that he had been successful in experiments to produce porcelain in the previous winter. By 1696 he was in discussions with the Saxon elector, Augustus II, concerning the building of glass and porcelain factories. The elector set out conditions that first Tschirnhaus find :-
... all places in Saxony with deposits of the precious stones jasper, agate, amethyst, and topaz.These would provide the raw materials needed for the factories and indicate the best places to site them. Glass factories were established in Dresden and Glücksburg around 1699 and their construction was supervised by Tschirnhaus. He made a trip to Holland in the winter of 1701-02, inspecting the ceramic factories in Delft and other places before putting his porcelain into production. From 1702 he worked with Johann Friedrich Böttger :-
... a con man and a jailbird with chemical experience and laboratory skills ...on the problem of producing porcelain. Tschirnhaus's wife Elisabeth had died in 1692 and in February 1704 he remarried, his second wife being Elisabeth von der Schulenburg zu Mühlbach. In 1706 Sweden invaded Saxony, forced Augustus to formally abdicate by signing the Treaty of Altranstädt in September. This put Tschirnhaus in considerable difficulty regarding his porcelain factories. However after the war he was offered the position of Chancellor at the University of Halle but remained on his family estate of Kieslingswalde. There was great competition from governments to obtain his porcelain techniques but Tschirnhaus kept them to himself and ended his life deeply in debt. Schönfeld writes :-
Tschirnhaus and Bottger fired the first cup of true unglazed porcelain in October . The cup consisted of a calcareous porcelain burned above 1,350?C (a temperature that produces a hard-paste porcelain). Its infusible ingredient was kaolin clay from Colditz, and its fusible ingredients were alabaster and calcium sulphate. Only days after this breakthrough, Tschirnhaus succumbed to dysentery and died. "Triumph! Victory!" were his last words.A factory at Meissen started production of his porcelain in 1710, one in Vienna in the following year, and the first sales of any consequence of Tschirnhaus's porcelain took place at the Leipzig Fair in 1713.
Hofmann gives this assessment of his contributions :-
During his university years he lacked the guidance of a kind, experienced, yet strict teacher, who could have restrained his exuberant temperament, moderate his excessive enthusiasm for Descartes' ideas, and instilled in him a greater measure of self-criticism. Even so, Tschirnhaus's achievements - often accomplished with insufficient means - were far more significant than the average contribution made by university teachers of science during his lifetime. Indeed, even his errors proved to be important and fruitful stimuli for other scientists.Although these critical comments are probably fair when directed towards Tschirnhaus's mathematical contributions, they are probably harsh when his philosophical contributions are considered where he did go beyond the ideas of Descartes.
Article by: J J O'Connor and E F Robertson