Using the Promptuary
by Jim Hansen
Those acquainted with Napier’s Bones will find the use of the Promptuary immediately familiar. The advantage of the Promptuary is that it reveals only those products needed for the solution and reduces multiplication to a simple procedure that anyone can follow.
The bones are more flexible, being able to multiply, divide and to compute both square and cube roots. Except for the sliderule, a spin-off invention from Napier's logarithms, no alternative portable computing device of that flexibility would be developed until the scientific calculator came on the scene in the mid-1970s, some 360 years after Napier died.
The tables inscribed on each of the computing rods of the bones differ considerably from those on the Promptuary slips, consisting of products from a single multiplication table. Promptuary tables carry the products of all digits 0 - 9 in a cleverly encoded matrix of 18 cells. Although Napier did not discuss how he arrived at this design, the encoding pattern was clearly described in his book Rabdology. I have given my account and interpretation here.
Principle of Operation
Promptuary operation reflects Napier's intent on making arithmeticic calculation into a procedure, but it still requires a pencil and paper. The calculation begins by laying out the numbers to be multiplied with the table and mask slips.
The table slips corresponding to the multiplicand are laid side by side vertical columns, with the most significant digit on the left as usual. The multiplier, designated by the mask slips, is laid horizontally on top of the table slips with the most significant digit in the top row.
The mask slips now show a clear diagonal line running through the center of each mask section forming what Napier termed "diagonal columns." In the following I will refer to the term "diagonal column" as simply "diagonal" so as not to confuse them with the more familiar "vertical column." The sum of the numbers displayed in each of the diagonals is a partial product of the solution. The result is found by simply adding up the numbers in each diagonal, forwarding any carries to the diagonal on the left as in ordinary arithmetic.
Multiplication Procedure
The calculation itself is probably best described as Napier intended, as a series of steps. Once the problem is arrayed on the Promptuary, perform the following steps, pencil and paper in hand:
Step 1. Starting with the left-most diagonal, write down the digit that is revealed. This is the least significant digit (lsd) of the product.Step 2. Add the numbers showing in the next diagnonal to the left and write down the least significant digit as the next digit of the product. Write down the most significant digit (msd) of the sum as a carry (if any) for the next diagonal.
Step 3. Repeat step 2 as many times as it takes to process all the diagonals.
Step 4. Write down the msd of the last diagonal sum as the next digit of the solution.
Step 5. If there is a carry from the last diagonal, write down this number as the msd of the solution.
Once all the columns have been processed, the final answer is complete. The Promptuary, according to Napier, should now be cleared and made ready for the next calculation by storing the slips in their rightful compartments.
A Demonstration
We will use a problem, the multiplication of 829 by 63 as a means to demonstrate and instruct on the use of the Promptuary.
Lay out the table slips for the multiplicand (829) next to each other, then the mask slips making up the multiplier (63). These are positioned horizontally on top of the table slips. Be sure to carefully line up the slips so the results are centered in the mask windows.
The diagonal columns are clearly marked. Notice that there are actually five diagonal columns in this example. These extend to the left of the vertical multiplicand strips as needed until every exposed number is accounted for.
At this point the solution can be determined. The procedure is this:
Step 1. The right-most diagonal always exposes a single digit, the lsd of the result. Write down the only number exposed in the first diagonal, in this case the number seven.
Calculations now progress to the next diagonal on the left, which exposes the digits 6, 2 and 4.
Step 2. Sum the digits in this diagonal (6 + 2 + 4 = 12). Write down the lsb of this sum ( 2 ) as the next digit of the solution, and carry its msd ( 1 ) to the next diagonal. We now know that the solution ends with the digits "27."
The digits found in each diagonal are summed with the carry from the previous column, if any. The lsd of the sum is written down as the next digit in the solution, and the msd, if any, is carried to the next column on the left.
Step 3. Repeat step 2 for the next digit. Here we add 4 + 5 + 2 and a carry of one from the last diagonal giving a total of 12. Writing down the 2 as the next solution number (it now reads "227"), we carry the 1 to the next diagonal.
Step 4. We are coming closer to the end of the calculation and so this diagonal is shorter than before. We sum up the numbers 2 + 8 + 1 and add the carry for a total of 12. Write the number 2 as the next digit in our result. It now reads "2227," and we carry the msd ( 1 ) to the next diagonal.
Step 5. The final diagonal has a single digit 4 in it. Adding the carry from the last column, we get 5. Write this down as the most significant digit of our result. Our answer, the product of 829 times 63 is 52227.
Always remember that if there is a carry in the last diagonal, it must be written down as the msd of the result.
Another Example
It takes just a little practice to quickly and correctly read out the diagonals. Try this example, already set up in the prototype Promptuary. The problem is to multiply 576 by 382.
With a pencil and paper at hand, locate the first diagonal which always contains the lsd of the product. In this case it is a 2, so on your paper, write it down.
The second diagonal contains a 4, 1 and 8, for a total of 13. Write the lsb of the total ( 3 ) down as the next digit of the solution, and carry the one to the next diagonal. Our solution digits are now 32.
Our third diagonal has a 0, 1, 6, 4, and 8, totalling 19, plus a carry of 1 for 20. The zero is our next solution digit, and we have a carry of two. The solution digits are now 032.
The fourth diagonal is the first of the shorter ones, and contains a 1, 0, 5, 1, and 1. These total 8, and adding our carry, we get 10. The next digit in our result is a 0, bringing it to 0032. Carry the one to the next column.
The fifth diagonal includes a 2, 5 and 4, giving a total of 11, plus the carry of 1 for 12. Write down the 2 as the next significant digit (our solution is now 20032), and remember the carry of one.
The sixth diagonal reveals a 4, 5 and a 2, plus our carry of 1 for a total of 12. Write down the number 2 (solution is now 220032) and remember the carry.
The seventh and final diagonal contains only the number 1 plus a carry for 2. Write down the number 2 and our solution for the product of 576 times 382 is 2220032. Happily, most modern electronic calculators agree with this answer!
Decimal Fractions
The Promptuary can perform multiplications using decimal fractions as long as you, the operator, keep track of the decimal point. This is no different than was required for the sliderule, which at one time your author used in both college and at work.
There is no difference at all, as far as the Promptuary is concerned, between multipling 1063 times 471 and 10.63 times 4.71. The product in raw number is 5000673 in both cases. It is up to the operator to adjust the answer to 500.0673 in the second case.
A Home for Promptuaries
I will agree with most who say that the Promptuary is a piece of mathematical nostalgia ready for the dark corners of museums. Or is it? It was designed by Napier as a means to allow anyone who can do basic addition to multiply very large numbers. Is there a population center still meeting these stipulations?
Yes, there is! In the lower grades where students are being taught the basics of arithmetic is an obvious example. The promptuary is a kinetic device that could very well serve both as an application of early arithmetic as well as a bridge to the next level for this age group. Perhaps sometime I'll post a lesson plan or two based on this most profound and magical device of the 17th Century.
Here's hoping that you've enjoyed the Promptuary Project!
Jim