How Slide Rules Work
Date: 10/11/95 at 12:31:34 From: Anonymous Subject: Napier's bones I am trying to complete an essay on John Napier's (Scottish Mathematician 15th, 16th century) contributions to math. One concept that I am having a hard time getting detailed information on is called Napier's Bones. He developed a method for multiplication using ivory rods with numbers marked off on them and by putting the appropriate rods beside each other he could read the answer. I'd like to find out how these rods (markings) were constructed and what math principles were used in determining the answers. Any sources that I have looked at don't reveal how he accomplished this. I have touched on his work with logarithms and have seen how you can solve multiplication of complex numbers by simply adding the logarithms of those numbers (same base) and wonder if Napier's Bones is just an extension of this concept. Thanks for any help! Stephen Powers, Fredericton High School Grade 10 NB Canada
Date: 10/11/95 at 16:42:0 From: Doctor Andrew Subject: Re: Napier's bones It sounds like Napier's Bones are an early slide rule. I'll explain a bit about how a slide rule works. You can probably verify that this is what Napier's Bones are. It is a property of logarithms that: log (ab) = log a + log b So by spacing numbers logarithmically on a bone or piece of wood, you can do multiplication. In fact, I can do it right here. I'll use logarithms in base 2 so I can deal with smaller numbers. What I've done is space the numbers so that the distance between two numbers is the difference of their logarithms in base 2 (log base 2 is written lg). So lg 2 - lg 1 = 1 - 0, lg 4 - lg 2 = 2 - 1 = 1, lg 8 - lg 4 = 3 - 2 = 1 etc. The spacing between other numbers, such as 3 and 2 isn't integral so I can't write them here without placing them wrong. All the numbers below are spaced evenly and are right justified. So here are my bones: 1 2 4 8 16 32 64 128 256 512 1 2 4 8 16 32 64 128 256 512 So now I want to multiply 8 by some number. So, I slide the 1 on the top bone to the 8 on the other bone like this: ------------> 1 2 4 8 16 32 64 128 256 512 1 2 4 8 16 32 64 128 256 512 So 1 * 8 = 8, 2 * 8 = 16, 4 * 8 = 32 and so on. Pretty cool, don't you think? Almost as good as a calculator. Now we really can write other numbers like 3, or 2.5 on these bones, but I can't position them correctly here. This is one way people did arithmetic quickly before calculators. I hope this helps. If you have any questions about logarithms please send them to us. I recommend looking up Napier in a Math history book or even an Encyclopedia, but you might have better luck looking up slide rule in a the Encyclopedia, or even finding a book about slide rules. Good luck! -Doctor Andrew, The Geometry Forum
Date: 2/24/97 at 14:05:14 From: Andrew Lister Subject: Re: Napier's bones I was searching for an image of "Napier's Bones" on the web, and came up with your answer to a 10th grader's query, but I fear that you didn't actually answer his question properly. However, as the original query was dated sometime in 1995, I'm a trifle late to help him. I don't know whether you are still interested, but my recollection (from a mathematics class in primary school in England, longer ago than I care to remember - Oh, OK, about 30 years) is as follows. Ideally, you need a graphic of this, but I don't have one ready, and I don't know how to send it to you anyhow. Let me know if you are at all interested, and I'll see waht I can do. In the meantime, I'll just have to try to describe them as best I can. Each "bone" has the proportions of a typical 12" ruler; say, 2cm wide by 22 cm long. This is etched with lines dividing it into a series of 11 squares. All but the first are also etched with a diagonal line dividing the square into two triangles. When the bone is held long side vertical, the diagonals run from top right to bottom left of each square. The first square (at the top or the bottom) has a single large digit. This will be one of the digits of the multiplicand. The squares below give the times table for that digit, writing the "tens" digit above the line and the "units" digit below the line. For example, the rod for 7 would read something like this: SEVEN 0 / 7 1 / 4 2 / 1 2 / 8 3 / 5 4 / 2 4 / 9 5 / 6 6 / 3 To multiply, select the rods corresponding to the digits of the multiplicand, and arrange them in normal decimal order. If you want to demonstrate this to kids, a great number to use is 12345679. Make sure that they are all lined up with tops and bottoms in line. Ask for the child's favourite number (between 1 and 9). Read off the result from the appropriate row of the bones. [It is convenient to have an extra bone to indicate the multiplier. This would be similar to the standard bones, but would not have the diagonals, and instead be marked with X 1 2 3 4 5 6 7 8 9 in large characters.] To actually read off the calculation, [this is the tough bit to describe] starting from the right, pairs of digits in the same diagonal stripe on adjacent rods are added. In other words, the digit above the diagonal (of the reading row) of one rod is added to the digit below the diagonal of the rod immediately to its left. If the sum is greater than ten, then it is necessary to carry 1 to the next sum. Sounds complicated, but anyone who can add two digits can easily learn to do it. (But you really have to draw it to see how...) To TRY to finish my example: 1 2 3 4 5 6 7 9 x7 0/7 1/4 2/1 2/8 3/5 4/2 4/9 6/3 0 7 4 1 8 5 2 9 3 +1 +2 +2 +3 +4 +4 +6 __ __ __ __ __ __ __ __ 8 6 3 1 9 6 5 3 carry +1 +1 __ __ __ __ __ __ __ __ 8 6 4 1 9 7 5 3 Funnily enough, this is the same answer I get with my calculator. Now, if you have followed me this far, and don't already know where I am going, you may be wondering why we got the child to pick a number. Well....depending upon their enthusiasm or patience, you get them to write down this intermediate result and now do it all over again, but this time just multiply by the "magic number" 3. Due to its special mathemagical properties this gives you, after some deft manipulation of the bones, er... 259259259. Hmmm, perhaps I forgot something. Or perhaps the magic is running at half power. Try again 259259259 x 3 = 777777777! Ah ha! I hope that this is of interest...
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.