Napier ruler
Napier's Bones (after John Napier , of this in his 1617 published work Rabdologiae seu numeratio per virgulas libri duo describes) are computing sticks with which multiplications and divisions can be performed. They are also called Nepersche stäbchen or Neperianische Rechenstäblein . The Arithmeum in Bonn, the world's largest museum of calculating machines, presents these calculators for multiplying. Around 1905, the Merkur Verlag Remig Rees company in Wehingen (Württemberg) produced this calculation aid under the name “Theutometer” on individual cardboard strips, which can be used for up to 18 digits.
The rods have a square cross-section. A row of the multiplication table is written in columns on each long side of a stick . For example, on the side of a stick shown on the right (Fig. 1) there are multiples of 7, from 1 × 7 to 9 × 7. At the top of each page is the respective basic number, in the example the 7.
Each number field is divided diagonally from bottom left to top right. In the lower right triangle there is the unit and in the upper left triangle the tens of the product. For example, in the 4th position of the rod of 7 there is 2 at the top left and 8 at the bottom right, corresponding to the product 4 × 7 = 28.
The sticks are placed on a tray for multiplication with the numbers 1 to 9 listed one below the other on the left edge. The chopsticks fit exactly into this tray so that they cannot slip vertically.
multiplication
The multiplication with the sticks should be explained using an example.
To calculate the product 7 × 46785399 , place the sticks on the tray according to the digits of the second factor so that on the far left there is a stick from the row of 4, i.e. with the number 4 at the top, and a stick from the Row 6, with the number 6 at the top, and so on to the last double crochet on the right with the number 9 at the top (see Figure 2).
The result is obtained from the digits in line 7 (which is highlighted in white in Fig. 2). Proceeding from right to left, read off the two digits within a parallelogram made up of two adjacent triangles , add them up and write down the units digit of the result. If the addition results in a number greater than 9, the tens digit (1) is included in the following parallelogram on the left. This creates the result of the multiplication from right to left, with the ones digit on the right, the tens on the left, and so on.
The result in our example is therefore 7 × 46785399 = 327497793 .
Multiplications with multi-digit numbers are also possible. To continue the previous example, the product 96431 × 46785399 (with the same second factor) should be calculated.
To do this, the sticks are also placed as shown in Fig. 2. The single multiplications with the numbers 1, 3, 4, 6, 9 (from right to left) of the first factor are then carried out one after the other and the results are written down one below the other, but each shifted by one place to the left, as shown in Fig. 3 .
Adding up the individual products gives the overall result of the multiplication 96431 × 46785399 = 4511562810969 .
Adding machine
After a reconstruction, Wilhelm Schickard used the chopsticks for his - the first - calculating machine in 1623, in which the addition of the partial products was carried out mechanically.
Web links
Individual evidence
- ↑ http://www.rechenwerkzeug.de/theutometer.htm
- ↑ The first "real" calculating machine. Retrieved March 13, 2019 .