The Tschirnhaus transformation, by which he removed certain intermediate terms from a given algebraic equation, is well-known. It was published in the scientific journal Acta Eruditorum in 1683.
In 1682, Von Tschirnhaus worked out the theory of catacaustics and showed that they were rectifiable. This was the second case in which the envelope of a moving line was determined. One of the catacaustics of a parabola still is known as Tschirnhausen cubic.
In 1696, Johann Bernoulli posed the problem of the brachystochrone to the readers of Acta Eruditorum. Tschirnhaus was one of only five mathematicians to submit a solution. Bernoulli published these contributions (including Tschirnhaus’) along with his own in the journal in May of the following year.
Von Tschirnhaus produced various types of lenses and mirrors, some of them are displayed in museums. He erected a large glass works in Saxony, where he constructed burning glasses of unusual perfection and carried on his experiments (1687-88).