Extracting Square Roots by means of the Napier rods

Welcome and have fun! To ensure, the HAFAL-factor (have-fun-and-learn) is maximum, first make yourself familiar with the basic usage in english / basic usage in german of the Napier rods by performing some simple multiplications. After that, continue here.

You can see a set of Napier rods (replica) here. You have to scroll down there.
The square root plate is a single piece that cannot be devided in any way. I just mention it, because it looks maybe like three 'strange' rods laid beside eachother. The same for the cube root plate.
The Napier rods "1" to "0" on the left handside are in deed all single rods. You can take each of them separatly away. But enough theory for now. Let's have some fun.

We are going to extract the square root of 5.805.690,25. Yes! And believe me, it's not the most difficult thing on planet earth. You'll see it soon. Ok, then ... let's go!

Before we start, let's already have a quick look at the final result of the calculation in the following figure A. Looks ..., hm?

Figure B gives a short help on how the rods look after each intermediate step. So we can check all the time that we are still on track.

figure 0: Start of calculation

Write down the number 5.805.690,25 and draw two horizontal lines below. Seperate the number by vertical lines into groups, containing each two digits. Start from the comma going to the left and to the right handside. See from figure 0 how it looks like.

figure 1: How it looks after Step1

Start with the very left group (group 1) of the number, which is 5 and compare it with the numbers on the left column of the square root plate (plate). Look there for the number nearest but smaller than, or exactly 5, which is 4 (second row). Write 2 (second row, right column) between the lines in the first group. This is the first digit of the solution!

Write now the 4 (second row, left column) under the lines in the first group (into L1). Subtract 5-4 (means H1-L1), which is 1, and write this 1 (the 'remainder') above the 5 (into H2).

Read the number from the plate (second row, middle column), which is 4. Therefore lay the 'rod 4' directly on the left handside of the plate. Check from figure B how the rods look like.

figure 2: How it looks after Step2

Write the 80 from group 2 (H1) up to H2, so that in H2 stands finally now 180.

Look from the rod and the left column of the plate for the number nearest but smaller than, or exactly 180. You'll find 176 (1/6+1/6) from the fourth row.

Write 4 (fourth row, right column) between the lines in the second group. This is the second digit of the solution!

Write now the 176 under the lines in the second group (into L2). Subtract 180-176 (means H2-L2), which is 4, and write this 4 (the 'remainder') above the 180 (into H3).

Read the number from the plate (fourth row, middle column), which is 8. Therefore insert the 'rod 8' between the 'rod 4' and the left handside of the plate. Check from figure B how the rods look like.

figure 3: How it looks after Step3

Write the 56 from group 3 (H1) up to H3, so that in H3 stands finally now 456.

Look from the rods and the left column of the plate for the number nearest but smaller than, or exactly 456. You will not find a smaller or exactly this number, because the smallest number is 481. In such a case write zero (0) between the lines into this group. This is the third digit of the solution! Keep the remainder (456) unchanged!

You cannot read any number from the plate in such a case. Therefore insert the 'rod 0' between the 'rod 8' and the left handside of the plate. Check from figure B how the rods look like.

figure 4: How it looks after Step4

Write the 90 from group 4 (H1) up to H3, so that in H3 stands finally now 45690.

Look from the rods and the left column of the plate for the number nearest but smaller than, or exactly 45690. You'll find 43281 (3/6+7/2/8/1) from the ninth row.

Write 9 (ninth row, right column) between the lines in this group. This is the fourth digit of the solution!

Write now the 43281 under the lines in this group (into L3). Subtract 45690-43281 (means H3-L3), which is 2409, and write this 2409 (the 'remainder') above the 45690 (into H4).

Read the number from the plate (ninth row, middle column), which is 18. In such a case, that this number contains two digits act like this. Insert the 'rod 8' between the 'rod 0' and the left handside of the plate. Now add the 1 to the rod left to the 'rod 8', which is 'rod 0'. As a result you'll get 0+1=1. Therfore remove the 'rod 0' and replace it by the 'rod 1'. Check from figure B how the rods look like.

figure 5: How it looks after Step5

Write the 25 from group 5 (H1) up to H4, so that in H4 stands finally now 240925.

Look from the rods and the left column of the plate for the number nearest but smaller than, or exactly 240925. You'll find exactly 240925 (2/4/0/5+5/2/5) from the fifth row.

Write 5 (fifth row, right column) between the lines in this group. This is the fifth digit of the solution!

Write now the 240925 under the lines in this group (into L4). Subtract 240925-240925 (means H4-L4), which is 0, and write this 0 (the 'remainder') above the 240925 (into H5).

When the 'remainder' is exactly zero (Hx-Lx=0), the exact solution for the problem is found! No more rods have to be inserted. The calulation ends here.

As long as the 'remainder' is greater than zero (Hx-Lx>0), the exact solution for the problem was not yet found. The calculation can be continued by adding more groups to the right handside of the number. Each of those groups contains '00' as value then. The more groups are added (means, the longer it will be continued calculating) the more accurate the solution will be.

I hope you had fun and could follow the explanations. If you have further questions, improvement proposals or you just want to tell me something ... feel absolutly free to email me to achim_behr@hotmail.com. If you'd like to create a link to this site on your homepage, just do so. I'd be pleased.

You can also extract square roots by only pencil and paper. Interested? - Look here.

You can do the same also with a Otis King Pocket Calculator or a Slide Rule, but not as accurate as with Napier rods or pencil & paper, ...but slightly faster.

Keywords:
Squareroot Square roots pencil paper Napier rods bones manual calculate multiplication division explanation slide rule sliderule / Quadratwurzel Wurzel ziehen Stift Bleistift Papier Napier Neper Rechenstäbchen Rechenstab Multiplikation Division Erklärung Rechenschieber