Napier's bones/temp

The Napier's bones article started off as a machine translation of an article from the Spanish Wikipedia. It needs lots of revision and editing before it is usable here. This is a work in progress.

This is a temporary work-page for working on this article.

---

Napier's bones are an abacus invented by John Napier for calculation of products and quotients of numbers. Also called Rabdologia (of Greek ραβδoς, rod and λóγoς, word). Napier published his invention of the rods in a work printed in Edinburgh at the end of 1617 also entitled Rabdologia. Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations of sum and division to subtractions. More advanced use of the rods can even be used to extract square roots.

The abacus consists of a board with a rim in which the Napier's rods will be placed to conduct the operations of multiplication or division. The board has its left edge divided into in 9 squares in which numbers 1 to 9 are written. The Napier's rods are strips of wood, metal or heavy cardboard. The surface of the rod is divided into 9 squares, and each square, except for the top one, is divided into two halves by a diagonal line. In the first square of each rod a single-digit number is written, and the other squares are filled with double, triple, quadruple and so on until the last square contains nine times the number written in the top square. The digits of each product are written one to each side of the diagonal and in those cases in which they are less than 10, they are written in the lower square, writing a zero in the top square. A set consists of 9 rods corresponding to digits 1 to 9. In the figure the rod 0 has been represented; although for obvious reasons it is not necessary for calculations.

Table of contents 1 Multiplication 2 Division 3 Square Root 4 Modifications

Multiplication

Provided with the described set, we suppose that we wished to calculate the product of number 46785399 by 7. In the board we will later place the corespondientes rods to the number, as it shows the figure, doing the reading of the result in the horizontal strip corresponding to the 7 of the square of the board, operation that only requires simple sums, with taken naturally of the digits located in diagonal.

Beginning by the right we will obtain the units (3), the tens (6+3=9), the hundreds (6+1=7), etc. If some digit of the number that we wished to multiply zero outside, would be enough to leave a hollow between the rods. Let us suppose that we want to multiply the previous number by 96.431; operating analogous to the previous case we will quickly obtain partial products of the number by 9, 6, 4, 3 and 1, placing them correctly and adding, we will obtain the total result.

Division

Also could be made once well-known divisions the 9 partial products of the dividend; determined these by means of the abacus, it is enough to select the inferior one immediately to the rest with no need to make the annoying rough estimates that require the made divisions by hand.

Square Root

As we know, to extract one firstly square root, must group the digits of two in two from the comma, as much towards the right as the left, being the number of the following form: ... xx xx xx xx , xx xx xx... Taking the pair (that could be only a digit) from the left (xx), number ' ' ' a ' is obtained ' ' finds out so that their square is equal or smaller than the pair. This will be the first number of the solution. Reducing of the pair the square of the whole number thus found, we obtain the rest: > = xx - a² (If the first pair were 07, the number to would be 2, and the rest 7-4=3) Later, and of iterative form, the following pair is added to the rest, being a number of the form yxx (and, the previous rest, xx the added pair) that we will call Ra. The following number of the solution will have so to be that the square of the partial solution ab (being ab a number of two digits, not a product) is minor who xxxx (both first pairs of being): > = (a·10 + b)² = (a·10)² + 2·a·10· + b² < xxxx clearing: + b² < xxxx - (a·10)² = R:(2·a·10 + b)·b < Ra (i) Operating equally once known the numbers ab, will have to determine the third number of the solution (c) and following (d, and...) that, as easily it is possible to be demonstrated operating to the previous case analogous, will have to fulfill: + c)·c < Rb (II):(2·(abc)·10 + d)·d < Rb (III):(2·(abcd)·10 + e)·e < Rb (IV):... The indicated products can be obtained easily with the abacus of Napier, but for it an auxiliary rod is necessary so that in each horizontal strip it gathers the squares of the corresponding numbers. Well-known first number ' ' ' a ' ' ', we placed in the abacus (or) the rods corresponding to duplo of a. Hecho this, will be enough to add the rod of the squares to find the number so that the equation is fulfilled (I), that will be corresponding to strip ' ' ' b ' ' '. the This number will have to be sustrar of Ra to find Rb. Found b, we retired the auxiliary rod of the squares and placed in the board the rod corresponding to 2·b; two cases can occur, if b is minor who 5, the double will have only one number with which it will bstará to place the rod; in opposite case (equal or greater than 5) duplo will be greater of 10, reason why it will be necessary to increase the last rod placed in a unit. Let us see it with an example. We wished to obtain the square root of number 46 78 53 99. We took the first pair (46) and we immediately determined the inferior square, that turns out to be 36 (49 that is the following one is greater than 46), so that the first number of the solution is 6, and the rest: 46 - 6·6 = 46 - 36 = 10. We placed the rods of 6·2 = 12 in the board, and next the auxiliary rod of the squares. We compose the rest and the following pair obtaining the number 1078 that will not have to be surpassed by the square of (6b). We read in the abacus (1) value 1024, finding that b = 8 and new rest 1078 - 1024 = 54, descending the following pair, we obtain a value of 5453. We placed the corespondientes rods to the double of 8; being 16 (> 10), we will retire the last rod, the one of the 2, replacing it by the one of the 3 (that is to say, we added a unit to him) and added the rod of the 6. The abacus is as it is in (à). As it can be observed, the placed numbers are the corresponding ones to the double of the solution found until the moment (68·2 = 136); that is to say, àbc.. of the previous equations.

Done this, we return to place the auxiliary rod, and operating as in the previous case, we obtain (2b) the third number: 3, being rest 1364. We descend the following pair obtaining a value 136499, we placed rod 6 (3·2) and found following digit 9 and rest 13478. While the rest is different from zero can be continued obtaining significant numbers. For example, to obtain the first decimal, we would lower pair 00 obtaining number 1347800 and would place the rods of 9·2 = 18, being left in the board the following ones: 1-3-6-7(6+1)-8-aid. Making the verification, the first decimal is obtained = 9.

Modifications

During century XIX [?], the neperiano abacus underwent a transformation to facilitate the reading. The rods began to make with an inclination of the order of 65º, so that the triangles that had to be added were aligned vertically. In this case, in each square of the rod slogan the unit to the right and the ten (or the zero) to the left.

The rods were made of way so that the vertical and horizontal engraving was more visible than the meetings between the rods, facilitating themselves much the reading when being the pair of components of each digit of the result in a rectangle. Thus, in the figure it is appraised immediately that: 4938271605 5 x = card Abacus. In addition to the previous abacus, Napier constructed a card abacus. Both, reunited in an only apparatus consituyen an historical, unique jewel in Europe that the Spanish National Archaeological Museum [?] has.

The apparatus is a magnificent box of wood with inlays of bone. In the superior part it contains the rabdológico abacus, whereas in the inferior one is the second abacus that consists of 300 stored cards of in 30 drawers with which 100 are covered with numbers and two hundred show small triangular drills that they allow to see solely certain numbers of cards of numbers when they are superposed those, so that thanks to the capable positioning of and others multiplications can be made until the amazing limit dede a number of 100 numbers by another one of 200.

In the doors of the box are in addition the first powers to the numbers digits, the coefficients of the terms of the first powers of binomial[?] and regular the numeric data of poliedros[?].

It is not known who was the author of this riquísima jewel, nor if it is of Spanish responsibility or it came from the foreigner, although it is probable that originally Academy of Mathematics belonged to the Spanish Academy of Mathematics [?] created by Felipe II[?] or that brought like gift Prince of Wales . The only thing that can make sure is that it was conserved in Palace, of where passed to National library [?] and later to the National Archaeological Museum, where still it is conserved. In 1876, the Spanish government sent the apparatus to the exhibition of scientific instruments celebrated in Kensington , where it called the attention extraordinarily, until the point of which several societies consulted to the Spanish representation about the origin and use of the apparatus, which motivated that D. Felipe Picatoste[?] wrote a monograph that later was sent to all the nations, surprising the fact that the abacus was only well-known in England, country of origin of its inventor.

Source: Hispano-American Encyclopedic Dictionary , Montaner i Simon (1887).