The mechanical calculating machine of Johann Helfreich Müller was built only a few years after Philipp Matthäus Hahn publicized his construction and was completed in a surprisingly short time. It was built by a watchmaker in Gießen according to the plans of the building engineer in Darmstadt and was demonstrated in 1784 at the Academy of Sciences in Göttingen. Like Hahn’s machine it is cylindrical and has stepped drums, but differed totally in many other respects: the fourteen setting wheels are turned by means of laterally placed serrated knobs, the entered digits appearing in windows below which there are short cylinders with inscribed numbers on their curved surface; the stepped drums are aligned horizontally; the tens-carry mechanism is technically new; and one can work with number systems other than to the base 10. The original machine (from the State Museum in Darmstadt) was disassembled completely by the Arithmeum technicians and cleaned, then every part (around 6000 pieces, 4000 of which were different) was measured, sketched and photographed. Finally, a mechanic of the Arithmeum wrote up a detailed description of how the machine functions. An exact, fully functional replica was then built according to these records. This took about three years. In order to make the replica look as much like the original as possible, the exterior surfaces were gilded. In the process of completing the replica, it had to be disassembled and assembled again several times. It weighs nearly 16 kg and is housed in a portable case which is also an exact copy of the original one.