by Jim Hansen
The world’s first practical calculator, one that could multiply, divide and find roots was developed in England during the latter part of the 16th century. John Napier's description of what we usually call “Napier’s Bones” comes to us through the book Rabdology (a term coined by him) or "Calculation with Rods." They are described in Part 1 of his 1617 book (written in Latin) Rabdologia. This was his last contribution to mathematics and was published in Latin and several other non-English translations following his death.
Napier's full dialog, artfully translated from Latin into easily understood modern English, is published by MIT Press as Book 15 of the Charles Babbage Institute's Reprint Series for the History of Computing. Information on this slim volume - some 135 pages - can be found at the publishers as Rabdology. All Napierian quotes used in this paper are taken from this tome, which was brought to my attention by the learned Ian Bruce, a prolific Latin translator of historical mathematics literature. For more links to Napier and his works than you could possibly ever review (and in a form more civilized than Google's), check out this site.
Napier is unquestionably one of the greatest of the early mathematicians. Working alone in the British equivalent of the "outback," at a time when the average person probably couldn't multiply 5 by 5, he developed what eventually became the foundational infrastructure of modern mathematics, with a keen understanding that arithmetic was the essential stepping stone to all greater mathematical understanding of the universe.
He was particularly devoted to - and probably first - in developing arithmetic calculation procedures that did not require an understanding of the mathematics behind the procedure. Or even what the calculations were for! In his time, this meant reducing multiplication, division and root calculations into simple addition and subtraction.
Early 17th Century arithmetic calculations left all remainders in fraction form because the decimal point had not yet been invented. Thus fractions that had to be carried forward complicated the process, making math calculation very cumbersome. On page 31 of Rabdology under "Excursus on Decimal Arithmetic," after describing division using his bones, Napier discusses what to do with the fractions left after a division.
Here he briefly describes the notation developed by "the learned Simon Stevin" in his Decimal Arithmetic and continues: "Perhaps the difficulty of working with fractions that have various denominators is not to your taste, and you prefer the ones whose denominators are always 10, 100, 1000, etc." According to Napier, Stevin describes what we would think of as ordinary long division, ending the whole number part of the result with a period. The numerals following the whole number was called "firsts", "seconds," and "thirds." Each decimal fraction number had ticks written above it to indicate the firsts, seconds, and so on. The image at the right shows this form.
Napier then comments that regarding the numbers following the decimal point, "there is the same facility as with whole numbers. If so, you can, after finishing the common division, end with a period...and add to the dividend or the remainder one cypher (Napier’s spelling for the word cipher) for tenths, two for hundredths and three for thousandths (or more if you wish)..."
Stevin did this, but added the order references above the fractional parts of the result. Napier dropped those references, and simply kept the decimal point. As Napier didn't cite anyone for its use except in combination with Stevin's notation, it might not be too much of a stretch to give him credit for inventing the decimal point as we use it today. Some go as far as to give him the entire credit for its invention. The use of the decimal point, in one form or another, was being discussed in Napier's time. It is unlikely that anyone can be positively credited with the invention, but it is clear that Napier was one of the earliest to have adopted and promoted its use.
No commentary on Napier would be complete without what most would agree was his greatest accomplishment: the invention of the logarithm. Although I have provided a brief Powerpoint presentation on this subject. No further mention of logarithms will be made here except to say that Henry Briggs, a friend of Napier, and who developed them into the form that we still use today, calculated his common log table of some 30,000+ numbers in fractions! He apparently never picked up on the utility of the decimal point. Logarithms where in common use for more than two centuries after Napier's death, and were taught as a subject in college math curricula until the late 1960s when the scientific calculator made them non-essential for most calculations.
Napier's place in history is secure. Probably the most detailed biography of him on the web is to be found in the 1911 Encyclopedia Britannica (Napier's Biography), which puts it this way: "...Mirifici Logarithmorum Canonis descriptio ... can be placed as second only to Newton's Principia."
The Bones have endured into the 21st century in ways limited only by our imagination. An interesting demonstration of bone use, written by Dr. Michael Caulfield and Wayne Anderson of Gannon University, can be found here or you can download a local copy from here. (You must have Flash and Java enabled on your computer.) Then of course, you could think about using my version of them, printed with a laser printer.
The calculator to beat for more than 400 years: Napier's Calculating Rods
The origin of Napier's rods are somewhat of a historical puzzle. He is generally given the entire credit for their invention, yet there are indications of their earlier use. First described in Rabdology, published after his death, they were already well known and available from a number of craftsmen and instrument makers. In his Rabdology dedication, he states: "I had two reasons for making my book about the manufacture and use of the rods available to the public. The first was that the rods have found favor with so many people that they could almost be said to be already in common use both at home and abroad. The other was that it was brought to my attention...that you had kindly advised me to publish them lest they be published under someone else's name..."
Without going into giddy historical detail, it must be said that the rods are but a clever use of what we call "the multiplication table." It was first described sometime before 572 BC by Pythagoras, the famous Greek mathematician. While Napier freely published how to make and use the rods, neither he nor anyone writing of them in later years discussed the mathematics behind them. Although we can easily see how the multiplication table and the rods work in concert, they are actually based on an ancient multiplication approach called the “quadrilateral.”
There isn't much written history on this rather arcane subject. This type of multiplication first appears in several Hindu works and appears later in Chinese arithmetic around 1593. In Arabia and Persia it was called the "method of the sieve" or "method of the net." In Italy during the 14th and 15th centuries it was called the gelosia or graticola method. In a 1478 book by Treviso, the illustrations shown there are dead ringers for Napier's rod layouts.
So perhaps diagonal multiplication and division were well known by Napier’s time. However, it is not at first obvious how his root process works. The approach he used is based what today we call the expansion of squares. An excellent monograph, describing this and the other mathematics behind the rods is given by D. J. Bryden in his Napier’s Bones: A History and Instruction Manual. This is a published work which so far hasn’t appeared in the public domain. Gareth Cronin in his The Works of John Napier describes some of the math processes, as well. You can find this paper here.
Napier indicated that the rods should be made of "silver, ivory, boxwood or some similar material." Ivory, which is somewhat bone-like in appearance, may have given the rods their common name, bones. Or it could be that the name implies a more philosophical meaning, perhaps, “the legacy of Napier.” In any event, the bones are little more than a set of the multiplication tables printed on square rods, sometimes as short as three inches in length.
The Napier's Bones Powerpoint presentation describes the bones and their use from an educational and classroom perspective. It uses the usual and profane uncredited web images, but can form the basis of an interesting short story or lesson plan. What follows is a bit more detail, plus an example of division and roots. I’ve adapted the bones to a paper form which makes them better suited for use in the classroom. If nothing else, you'll find that the bones are a fascinating adaption and clever use of what used to be taught as the multiplication table.
The bones consists of a "plate" and a set of "rods." Napier described three plates: one for multiplication and division, one for square roots (quadrata), and one for cube roots (cubica). Once the rods representing a number are placed on the plate, only simple addition or subtraction is required to solve any problem posed. My paper version of the bones is shown here and is available in PDF format. It was painfully produced using WordPerfect because it has a diagonal feature in its table function. A square root plate is included in this file, but I was too lazy to make up a plate for the cube root.
As shown in the image to the left, the multiplication and division plate is simply the numbers 1 through 9 written in the left-most vertical column. In this set of bones, the plate is glued on the left side of the case and is identical to the one rod. The ten rods are labeled at the top with the numbers 0 through 9. As each rod has four sides, each inscribed with a different number, allowing calculations with some numbers having duplicated digits.
The length of the rods is divided into ten rows, each with a diagonal line scribed as shown. The numbers in each row are the product of the rod number multiplied by the row number read from the plate. In the example at the left, for instance, each rod entry is three times the row number. The diagonal line separates the most significant digit (msd) and the least signficant digit (lsd) of the products.
Constructing a Set of Bones
I teach the Bones following substantial drill using a multiplication table. For student use, I print the Bones on two different colors of paper. One color is used as the plate (in these examples, blue). The other color (white) is cut into columns that serve as the rods. I sometimes give students two or three "rod" pages, to allow computation of numbers with repeated digits, such as "2266."
A full-sized set of bones were developed for classroom use over a number months in hit and miss sessions. These are all available for download as described in the following text. Another nice set which includes the plates for both square and cube roots were found on a most interesting site at site where it appears with copies of Genaille's Rods. There is no commentary on this site, just these images. (Genaille-Lucas Rods as they are sometimes called, were the only important improvement made to the Bones, but came on the scene very late, around the turn of the 20th century, and were soon abandoned as mechanical calculators became common.)
The illustration on the right shows the number 3579 set up on the multiplication/division plate. This number can be multiplied by any number chosen, although it is always best, to minimize your addition, to set the larger of the two numbers being multiplied using the rods. In this case the number to be multiplied (3579) is found on the plate, as we’ll soon see, digit by digit.
A number to be multiplied is set up by the rods and each digit of its multiplier is selected from the plate. In other words, the digits on the plate produce multiples of the value shown on the rods. The product is read out after the values shown in the diagonals are added together. Looking at the ONE row on the plate, we see that 1 x 3579 is, indeed, 3579 as read out across that row. Looking at row two on the place, we find that twice 3579 is 7158. This answer is read out only after adding the values in the diagonals on row 2 of the plate. (I have highlighted alternate diagonals for easier reference.)
The first diagonal column (the right-most diagonal under the nine rod) holds an eight , and so the least significant digit is eight. Writing this down, then the added diagonals going left to right, gives the solution as 0, 7, 1, 5, 8 or 7158.
Another example: looking at four times 3579, go down to the fourth row of the plate and, summing each diagonal column shown on the rods as before, we find the answer is 1, 4, 2, 11, 6. The second addition summed up to 11, and so we must carry one to the next column, making the result 1, 4, 3, 1, 6 or 14316. Notice that this approach is somewhat different than the one described in the Powerpoint presentation. Use whichever seems most appropriate to you.
Remember: if the sum of a diagonal is over 9, such as "8+3 = 11" write down the least significant number as the answer for that column and carry the most significant digit to the next diagonal column on the left. And if there is no column to the left, that carry then becomes the msd of the product.
Multiple digit multipliers are processed the same way, digit by digit. These results must be written down and shifted a column just as in ordinary pencil and paper arithmetic. Once all the numbers have been multiplied, the partial sums are added to find the result. In this way the bones eliminate the need to know or understand multiplication tables entirely because multiplication has been replaced by a simple additive procedure.
Multiplication can start with either the most or, as is commonly done in today’s hand multiplication, with the least significant digit. The only difference is in which direction the partial results are shifted. For this description, we'll multiply our 3579 by 456.
The basic technique is simple: go to the sixth column and write down the product of six times 3579. Next go to the fifth column and, shifting to the left one column, write the result of five times 3579 underneath the previous result. Notice that this is exactly the same thing we do with pencil and paper. Now go to the fourth column and, shifting to the left one column from the last result, write down the result of four time 3579. Draw a line under this result and add all the columns together.
Your results should look like this:
The number you are multiplying by 456: 3579 The results of 6 times 3579: 21474 The results of 5 times 3579, shifted left one place: 178950 The results of 4 times 3579, shifted left two places: 1431600 The total, the three partial sums added together: 1632024
Bone division is slightly more complicated than multiplication. We show the process using the same setup, with divisor presented by the rods. Division is performed by finding the largest multiple of the divisor that can be subtracted from the left side of the dividend, taking that number as a digit in the quotient (which Napier termed "quotumus"), and performing the subtraction. It is a process very similar to normal division except, again, no knowledge of the multiplication table is required and only simple addition and subtraction is involved in the calculation.
By way of example we will divide 3579 by 42 (see the image at left) using the rods and original pen and paper technique described by Napier. Napier was detailed in his description of the process and prescribed an organized way to keep track of computations. After the rods are set up, on a piece of paper, write down the dividend (3579) followed by an open parenthesis. The row number for each partial solution represents the quotumus, which is written as it is discovered to the right of the parenthesis.
Beginning the calculation, remember that the row values under the two rods (the four and five) indicate multiples of 42. For example, row one gives the original value, 42, but row two doubles this value to 84, and so on. We will look for the largest multiple that is equal to or closest to our divisor, 42. The row number providing this value is the quotumus and becomes the msd of our quotient.
Looking down the columns, at row eight we find values of 32 and 16 which, when added diagonally, become 320 plus 16 or 336. Row nine yields 360 plus 18, or 378, which is over the value we're looking for, i.e., 357. The most significant digit of our quotient is therefore 8. Write the value (336) under the dividend, and the number 8 to the right of the parenthesis. Now subtract 336 from 357 and write the result (21) above the dividend. This completes the first iteration of the division process.
The next iteration starts by imagining the next digit of the dividend being placed after the remainder, making 219. We now examine the columns for a number equal to or just under 219, and find this value, 210, is contained in row five. Write the quotumus to the right of the 8 following the open parenthesis, the number 210 offset one digit to the right and under the 336. Subtract 210 from 219 to find the remainder. The remainder, 9, is now written above the previous remainder, offset one digit to the right.
At this point the problem is solved as far as most in the early 1600s would have carried it. The solution to 3579 divided by 42 is 85 with a remainder of 9 - which would have been kept as a fraction of 42, or 85 9/42, or in reduced form, 85 3/14.
As mentioned earlier, Napier advocated placing a decimal point after the whole portion of the quotient and continuing with the division, just as we would today, by placing a few zeros to the right of the original dividend, and a decimal place to the right of the quotient. This is an exercise that will be left for you to complete; the result is 85.2142857. If you happen to get a problem that divides evenly, the remainder placed on top of the number stack will be zero, indicating to the 16th century math whiz that the process is complete.
It is amazing how quickly division can be done with the Bones, and doing just three or four examples will give you a lot of confidence. Remember, this is just a procedure, one that in the case of division uses a relatively simple table lookup, a comparison and a subtraction. Although the bones always requires these steps for each quotient digit, there is no use of the multiplication table required. You will probably find root extraction a bit more obtuse at first, but working out a few problems on your own and comparing the results with a calculator shows that you can depend on the system.
The Bones can be used alone to extract square or cube roots, but this would require separate calculation and subtraction for each multiple of the divisor and the square or cube. Napier foresaw this problem and designed a plate that avoids some of this work when finding square and cube roots. The square root plate provides three columns that are positioned identically with the rods. Only those who have learned the arcane art of root extraction can fully appreciate Napier's invention. I'll go through the square root process, but because I doubt many are all that interested, Napier's cube root procedure and plate is not covered.
It must be remembered that this is a manual process, one that requires pencil, paper and thought. The rods will provide the multiplication, but you still must be able to perform diagonal addition and subtraction. The process is quite similar to division on the rods, but you’ll need to do several problems, checking them against your calculator - before a comfortable feel for the process develops. Exercises are easy to generate - just enter a number in your calculator, write it down as the answer, then square it. Then use the rods on the square to find the original number.
The square root plate consists of three columns of numbers. The right-most column provides number N (1 - 9). The center column contains the value of 2N. And the left-most column provides values of N2 with diagonals separating the left (most significant) digit. This plate (see image at right), the rods, a pencil and some paper are all that you need to calculate square roots of numbers of any size. Once you get past having to look at the instructions or making mistakes, root extraction is only a little slower than ordinary division with the Bones.
Napier's only square root extraction example operated on 117,716,237,694, a rather large number. I checked the results with my TI-84 which showed the answer is about 343098.0001; the TI-89 refines it two more decimal places to 343098.000131. The Bones say that the answer is 343098 with a remainder of 90. So who are you going to believe? Hint: If accuracy is of extreme importance, don't always trust your calculator’s floating point package - they are not always right! Do the extraction by hand as a check!)
We will use the rods to find the square root of 4305625, a big enough number to avoid an obvious root. Write the number down, separating pairs of numbers starting on the left end of the number. Napier placed periods between the pairs, but since they can be confused with decimal places, we will put a raised period between each pair, and a decimal point at the right end of the number. You should thus have dots between these pairs of numbers: 25, 56, and 30, plus a decimal point. We thus have three periods and a decimal point (four in all), and this tells us that the root will have four whole numbers.
Draw a horizontal line under your number, leaving enough space to write the root between the number and line. Focus on the first number, the four, then look at the numbers in the left column (N2) of the plate. Find one whose diagonal sum is equal or just smaller than four. A four is found in row 2 of the plate, making the quotumus two. Write this number between the horizontal lines under the first dot.
Subtract the four (from the plate for row 2) from our first number (4) and write the remainder (0) below the second line. Place a rod whose number is double the quotumus value just determined (2) next to the left side of the plate. (If you have trouble doubling numbers, use the 2N column on the plate to find it!) Bring down the next pair of numbers (30) and write them following the remainder.
We are now ready to repeat the process for the next digit in the solution. Start by looking at the left-most pair of numbers (30) then look down the column of squares and the number 4 rod next to it. This time we are looking for a diagonal sum equal to or just less than 30. Suddenly we find a problem - the first diagonal sum (for row 1) is 41, and all values below that are greater.
When this happens, it means that the quotumus is 0. Write a zero following the previous quotumus of 2, insert a zero rod between the 4 rod and the plate, and bring down the next two numbers (56), appending them to the 30 already there from the last iteration. This gives us 3056 to work with. Next, scan the rods and left side of the plate looking for a value equal to or just less than 3056. The closest value is found at row 7, making 7 our next quotumus (write it in the answer field) and the value, 2849 is subtracted from our 3056. This gives a remainder of 207.
At this point our answer is 207 and we’re ready for the next digit. Bring down the last two digits (25) and place them after the 207 to give 20725. Take the last quotumus (7) and double it to give 14. In our previous iterations this gave us a number below 9 which was used to choose a rod. There is no rod 14, and so what we do is add the 1 to whatever the last used rod value was (the zero rod in this case), and replace that rod with the new value rod whose value is the sum of the old rod and the msd of the number. In this case, we add 0 (the old rod) to the 1 of our number 14, and replace the zero rod with a one rod. We then place a four rod (the last digit of 14) next to the plate, making our rod stack to the left of the plate now 414.
Proceed as before, this time looking for a diagonal sum that is equal to or a little less than 20725. And as if by magic, it appears exactly on line 5, our new quotumus value. And so we have a perfect root: the square root of 430565 is exactly 2075. Who would have guessed? As you can see, the square root process is really almost the same process as division.
Pick some numbers of your own and practice with the method - it is really quite interesting how it works out so simply. Cube roots end up being no more difficult to do with the Bones, but that’s a story for another day.
How it Works
But what is actually going on here? How does it work, mathematically? Using today’s algebra, it really isn’t all that difficult. The process assumes that the number we want to find the root for is an “expanded square” and eventually discovers the answer through successive approximation.
For our purposes, all numbers can be expressed as the sum of two numbers. And so if we took two numbers and added them together, it could be written as (a + b), right? And if we took (a + b) and multiplied it by itself, we would be squaring the sum of a and b, which is just a number. At this point we can say that (a + b) is the square root of (a + b)(a + b).
With this in mind, let’s go looking for the square root of 28900. We’ll start by expanding (a + b)(a + b) (multiply these two terms) to give a2 + 2ab + b2. Because we are looking for the square root of 28900, we could set a2 + 2ab + b2 = 28900. We could find the root of 28900 by simply guessing a value for a, then calculating what b would have to be for the equation to equal 28900. The result would show us that another guess or “adjustment” is needed to give a closer result. This process would have to be repeated many times if wild, unorganized guessing were used.
But an organized approach to the problem gives much quicker - and better - results. Let’s separate our square number into two-digit tokens starting at the right side of the lsd. The value of the square root will be placed above each separator. Napier used decimal points as separators which worked fine before the decimal point came into common use. As we might confuse the decimal point markers with the real decimal point, we’ll raise the markers so they’re in the center of the number we’re working on as shown to the right. We’ll need three markers, one on the right end of the number, another between the 9 and two zeros, and a third between the 2 and 8. Since we have three markers, that means the square root of 28900 will have three digits to the left of the decimal place.
We need to find an approximate number that when squared does not exceed 28,900. This is our first “guess” for the value of a. This number, because it is to the left of the decimal point by three places (or three separators), it represents the hundreds digit in the square root. If we square 100, we get 10,000; if we square 200, the answer is 40,000, so we know the square root of 28900 is somewhere between 100 and 200. Using either the squares plate or by examination, what number, when squared, is equal to or less than two? This number, 1, appears under the N2 column of the squares plate in row 1. Subtract this number (1), the quotumus, from the 2, and write down the answer (1) and bring down the next two digits, the 89 making 189 the next number to be worked on. This is shown on the left.This completes our first step. What we’ve done is guessed that 100 squared (10,000) is close to, but less than 28,900, which it is. We then subtracted this 10,000 from 28,900 leaving 18,900. The following steps now improve our original guess by filling in digits with numbers less than 100. We’ll next be looking for a square value less than 18,900. This is a reiterative process and will be repeated until the exact square root is found or you’ve reached an answer with the decimal precision that you want.
We need to make a new guess: what number when squared will be equal to or less than 189. The squares template is a big help here, but we need to add the next rod to it so values in the range of this number will show up. The rod is selected by doubling the quotumus just found. Doubling the 1 gives 2, and so we set up rod 2 to the left of the plate. Looking down the 2 rod and N2 columns, and adding diagonally, we look for the largest number less than or equal to 189. And we find this value, 189 at row 7 on the template. Write 7, the quotumus, over the second separator, and the square value, 189, under the 189. Subtract the 189 from the 189 to give a result of zero. Bring down the next pair of digits (00).
Our answer is 17 with one digit to go. Since there are nothing but zeros to be worked with below, we must come up with a number that, when squared, equals zero. That number would be zero, and so this is the last digit that we’re looking for. Write it as the last digit, above the decimal point. Using your calculator, multiply 170 by 170 to give 28,900. You might confirm this by taking the square root of 28,900 as well.
Wrapping it All Up
I hope you’ve enjoyed working with Napier’s Bones. They give a great sense of continuity between numbers and calculation. When possible, you might consider offering your students an even more authentic experience by using feather pens when working with them. It’s fun and the kids from other classes are often envious of the pens.
Writing with feather pens or their replacements, the steel nib and eventually the fountain pen, is much more difficult than with modern pencils and ball point pens. In fact, the first generation of ball point pen users, those of us who were in school in the 1960s, probably don’t even remember “fountain pens” which personally, I was never able to make work properly. I do remember my school desks of the time all had that mysterious hole in them where the ink wells were kept.
BIC pen, the first practical ball point pen, made writing enormously easier. So easy, in fact, that public schools essentially stopped supporting penmanship classes, once the age of typewriters and word processors arrived a few years later. Since then we’ve all been trying to read each other’s writing.
Enjoy playing with Napier’s Bones!