Talk:Napier's bones
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moved temporary article out of main namespace
Note: you should modify "game of rods" with "set of rods" -- wrong translation of Spanish "juego". --216.177.2.208 13:00, 17 Jun 2004 (UTC)
I revised the "Division" section. Someone want to check it for accuracy? There was essentially no info after translation to work from, so I had to follow the diagram. Tigerattack 19:50, 23 Jun 2004 (UTC)
Napier's bones are three dimensional, square in cross section, with four different rods engraved on each one. A set of such bones might be enclosed in a convenient carrying case. 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. Posted by User:203.131.154.109
Is that addition picture correct? It starts out adding the two digits on the diagonal (staring on the right) but somewhere in the middle starts duing something else. What? Rmhermen 17:08, Aug 2, 2004 (UTC)
- The original explanation omitted to mention carry-over, which I have now included. -- Picapica 08:40, 29 Mar 2005 (UTC)
Contents |
[edit] Some random info
Napier termed the so-called card abacus a promptuary, so I'd like to suggest this word may be a more accurate title for that section.
For what it's worth, a pdf scan of the original Rabdologiæ can be found archived here at the Computer History Museum.
My understanding is that this book has basically three interesting ideas -- the bones, the promptuary, and a trick to do binary arithmetic (including multiplication and division.)
The binary arithmetic stuff might be nice to include in this entry, there's an explanation of some of the operations at http://courses.cs.vt.edu/~cs1104/Napier/Chessboard.html
Best regards, -kb
[edit] Division error
There's a mistake on the line for the "multiply by 7" term.
It should be 771448 rather than 761448.
With that correction, the rest of the calculation comes out to be 485 with a remainder of 16364. The method provides (the exact) quotient and remainder, so the note about "...not the exact value..." etc shouldn't be there, and the solution could be be written as 
.
Best regards, -kb
I went ahead and corrected the arithmetic and added a short paragraph showing how you can find a decimal answer instead of fractions. Also, changed a label in the first diagram to "set of rods" per the first posters suggestion.
I couldn't find a good match for the existing fonts, so if someone can put up a more consistent picture, that'd be great.
--Kbs 16:52, 17 Sep 2004 (UTC)
[edit] Square root explanation
I'm starting to rewrite the section on square roots, you can see my current efforts at my user page. I've decided to redo it from scratch, for one thing I think it is simpler to start with the technique and add the theory later. Also, the original bones actually have three columns on the square root bone, and that isn't clear from the current images.
Please let me know what you think at my talk page.
--Kbs 08:43, 30 Oct 2004 (UTC)
I've now rewritten the section on square roots, I hope it is more useful than what used to be there!
It would be great if someone is able to make images for the various boards to make the article look more consistent. In any case, this perl code is what I use to generate the tables. I add a few attributes to its output as needed to customize the table.
my @x = (1,3,6,7,9,8);
print '{| border="1" cellpadding="2" cellspacing="0" align=left';
print "\n! ";
for my $i (@x) {
print " || $i";
}
print "|| √ || \n";
for (my $n=1; $n<=9; $n++) {
my $rtot = 0;
print "|- align=right\n| '''$n'''";
for my $i (@x) {
my $d0 = $i*$n%10;
my $d1 = ($i*$n - $d0)/10;
print " || <sup>$d1</sup>/<sub>$d0</sub>";
$rtot = $rtot*10+$i*$n;
}
my $d0 = $n*$n%10;
my $d1 = ($n*$n - $d0)/10;
my $d2 = $n*2;
if ($d2 < 10) {
$d2 = " $d2";
}
$rtot = $rtot*10+$n*$n;
print " || <sup>$d1</sup>/<sub>$d0</sub> $d2 $n";
#print " || ''$rtot''"; #uncomment to get the value column
print "\n";
}
print "|}\n";
--Kbs 22:45, 5 Nov 2004 (UTC)
[edit] Updates from the spanish entry
Guillermo just pointed out to me that he's been updating the spanish entry on Napier's bones.
It contains a new section for cube roots (which I'd started to write as well) and it also has corrected images for the division step as well as images for the four sides of the rods, and so on.
Unfortunately, I don't understand Spanish, but from the little I could follow from his pictures and equations, his edits look very useful, and moreover it will stay consistent with the rest of the entry since everything else comes from there anyway.
I'll leave in my efforts for now, but my vote is that if or when someone does translate his entry, it be used to replace the square root (and maybe the cube root if I get around to it) sections.
--Kbs 03:55, 8 Nov 2004 (UTC)
[edit] Parent article: Rabdology
This article is getting a bit big, and I'm hoping that eventually there will be more information on the promptuary (called here the card abacus) and location arithmetic from Rabdology.
It doesn't seem right to put all this in the same entry, so I've started a parent page on Rabdology and created Template:Rabdology with links to Napier's bones and stubs for separate entries for the promptuary and location arithmetic.
Please let me know if this is nuts or reasonable. If noone objects and the promptuary entry does get expanded, I'll remove the card abacus section from here.
--Kbs 07:42, 25 Nov 2004 (UTC)
[edit] Pen & paper?
How is this method better than just using pen & paper? The latter seems much easier to me. 82.139.85.33 20:42, 20 August 2006 (UTC)
- At least for multiplication, you seem to have a point (it's not much harder to just do place-by-place mutiplication by hand). On the other hand, while I don't know about calling them "better" (that's such a subjective word), the bones do make things like division, or especially calculation of square or cube roots, much simpler from a pure grunge-work standpoint. This seems to be drifting a little offtopic (the article is about a technique that, for better or worse, was in fact used), but I hope it answers your question. —Ryan McDaniel 22:53, 30 September 2006 (UTC)
[edit] two digit multiplication
How would one solve a two digit multiplication equation on the Napier's bones. For exampl; 30 x 367!?
[edit] Napier's Abacus
I've been reading Knotted Doughnuts and Other Mathematical Entertainments (1986) by Martin Gardner. In it he describes an abacus by Napier that's related to moving pieces on a chessboard rather than the card based one mentioned in this article. He calls it one of the earliest binary computers (though Napier didn't realize at the time apparently)
Is this the right place to collect that info? —Preceding unsigned comment added by Lot49a (talk • contribs) 18:53, 24 April 2008 (UTC)
There's already a separate entry for this -- see Location_Arithmetic which is also referenced at the head of this article under the "computing devices in Rabdology" block.
