Step 13: Cubes and Cube Roots

Notice the K scale at the very bottom of the virtual slide rule. It is three copies of the C and D scales made smaller and laid end to end. It can be used for finding the cube or the cube root of any number. 

Finding the cube of a number is very much like finding the square of a number. Find the number on the D scale. Find its cube on the K scale. See the first graphic. By now you can recognize the C and D scales, even though they are not labeled. The hairline is over the 3 on the C and on the D scales. With you eye follow the hairline downward to the K scale, which is the very bottom scale. Notice that the hairline crosses over the number 27 on the K scale. The cube of 3 is 27. Look at the 4 on the C and D scales. If you look down to the K scale, you can see 64 is directly below it. The cube of 4 is 64. Look at the 5 on the C and D scales. Look down to the K scale. The number below it is 125. The cube of 5 is 125. You can do the same with 6 on the C and D scales. Below (if you could read it well) is 216. The cube of 6 is 216. (Although you cannot accurately read the 6 in 216 on the K scale, you know that 6 x 6 = 36. The last digit in 36 is 6. Again 6 x 6 results in a number that ends in 6. So, even though you cannot read it accurately from the slide rule, you know the last digit in the cube of 6 is also a 6. You can accurately read 21. Add the 6 you know must be part of the number and you have 216. This is a way you can frequently determine a digit that is beyond what you can read accurately on the slide rule.)

Finding the cube root of a number is more complicated than finding the cube of a number. See the second graphic. Basically, you mark off the digits in any number into clusters of three beginning at the decimal point. So, 1,200 would be marked off as 1 + 200. After groupings of three have been marked off, pay attention only to what is remaining. If there is one digit remaining, use the far left third of the K scale. The setup on the slide rule for finding the cube root of 1,200 is shown in the first graphic. Notice the hairline is set at 1,200 on the K scale. Read the answer on the D scale. The numbers on the D scale are 106 plus a tiny bit more. An electronic scientific calculator indicates that the cube root of 1,200 is 10.627.

Find the cube root of 12,000. The setup would be the same, except that the middle of the three sections in the K scale would be used. This is because two digits are left after removing groups of three digits. The digits one can read on the D scale are 229. The electronic scientific calculator indicates the cube root of 12,000 is 22.894.

Find the cube root of 120,000. After removing the first cluster of three digits, three digits remain. Use the far right segment of the K scale. The numbers indicated on the D scale are 494. Checking an electronic scientific calculator, the exact cube root of 120,000 is 49.324. 

There are rules for placing the decimal point when calculating cube roots. They are somewhat involved. In the last step I will link a couple of manuals so that those who wish to become very proficient at cubes and cube roots can learn the exact rules. Or, you can do some guessing in your head and know where to place the decimal point. For example, when calculating the cube root of 12,000 above you know the significant digits are 229. You could guess the number is about 20 just for a test. 20 x 20 is 400. 20 x 400 is 8,000. That is close enough to 12,000 that you now know where to place the decimal point. 

The process of calculating the cube root of a number smaller than 1 also has its special rules. Rather than bog down this Instructable with them, I would refer you to a manual I will link in the last step. 
<p><em>&quot;Most slide rules include a C1 scale&quot; ... and in other part of text.</em></p><p>I think is not a C1 scale but a CI (inverted) scale.</p>
<p>You are correct. I misread the manual when I got my first slide rule more than 50 years ago and have been making that mistake consistently ever since. I have made corrections, also where I had D1 rather than DI.</p>
<p>Had to use a slide rule back in high school for a drafting class. Darn if I have forgotten how to use it. Great tool for when the power goes out. Kids of today do not know what they are missing.</p>
I didn't get to read on within your instructable for some time, but today I did. I think I understood how to work the basic calculations that can be done, and then I wondered where to find real slide rules. <br>I found eBay and such, but I'd rather get a new one (I got interested in slide rules first about 2-3 years ago, but then lost interest because I couldn't get one and I don't like buying from eBay). <br>Today I've been looking around the internet and actually found the web site of Faber Castell (a German company) again. I read about slide rules there when I first got interested, but they didn't sell them. <br>Today, I actually found they started selling them again. So if any one is interested, you can look up all sorts of slide rules here that are for sale: <br> <br>http://service.de.faber-castell-shop.com/Rechenstaebe/Faber-Castell-Rechenstaebe <br> <br>And, with smartphones all around, you can find one that looks good for Android for free on Google play ;-) <br>https://play.google.com/store/apps/details?id=com.timscott.sliderule&amp;feature=search_result#?t=W10. <br>(I'll download mine tonight)
You are in a fortunate situation. Faber-Castell is a German company and you live in Germany, so you do not need to worry about arranging international shipping. It appears Faber-Castell is selling new old stock, that is, slide rules they produced before they ceased production in 1973, but never sold. On the one hand, these slide rules are almost 40 years old, but are also new because they have never been used. Thank you for your comment. I do not have a smart phone, but the slide rule application is interesting. There are also various other firms that sell slide rules, most of them used.
Yep, went all the way through college using a slide rule. Where I worked, the recent engineering hires had never seen one. After I graduated (1966) I took the first of my PE exams using a slide rule, but others brought those big mechanical/electrical calculating machines that were bigger than a typewriter. They had to arrive early to get a table near an electrical outlet. <br> <br>Bill
I almost wish I had gotten some training and experience using the log-log scales. As much as I enjoy using a slide rule, I have to find an excuse to do it. When a slide rule is nowhere near, I use the basic calculator on my cell phone. <br> <br>Thank you for looking. One day we old guys who can still use a slide rule will be appreciated.
Awesome. And I hope you go big, and go Christian for #200.
When I began submitting Instructables I thought I would run out of ideas after 25 or so. It was a shock to reach 50. But, things keep breaking around our house, or an idea comes to me that I want to share. Thanks.
...and to know that they have sent men to the moon with this thing!<br>Nice I'ble.
Thank you for looking and for your comment. The moon shots did include a primitive on-board electronic computer, but a huge amount of the engineering and any in flight human calculations would have used slide rules. In my sample problem of 259 x 653 the difference between the slide rule answer and the electronic calculator answer was 127 out of 169127. That is less than 1/10 of one percent. That is like calculating something to be at a distance of 1km with an error rate less than 1m. In a previous Instructable that I linked I summarized some points from a <a href="https://www.instructables.com/id/Refurbish-an-Old-Slide-Rule/" rel="nofollow">2006 article on the contribution of slide rules</a>&nbsp;(the final step). Things designed with a slide rule were a little over-engineered, but the slide rule was very effective. A final step in any engineering project was to check all calculations for mistakes.&nbsp;
Another great Ible from you my friend, thank you. <br>Also I would like to wish to you and your family a Happy Healthy and Prosperus New Year
Thank you, Steli. I wish you a prosperous and healthy New Year, too.
Phil, this is very valuable information, I want to go out and buy a slide rule.<br><br>I don't read all the text (it is late night). You can calculate also X**(2/3) and X**(3/2) with a cheap slide rule.
Thank you, Osvaldo. Ask around. Someone probably has a rule they would sell or give you. I thought you still had a slide rule.
Think Geek sells a slide rule that is very inexpensive. Bearing in mind that you get what you pay for it might be a good introduction. I had to take mine apart and clean it to make it slide anything like usable. At first I thought it was glued in place.
Thank you. I did link the ThinkGeek rule. But, because a number of reviewers had the difficulty you describe, I suggested interested people might want to check eBay for a comparable never sold plastic Pickett rule.
WOW. This looks interesting and complete. I have put this on a mental list of things to read and understand this month. <br><br>James
James, it is pretty complete, except for the material on logarithms and material related to placement of decimal points with square and cube roots, as well as material on square roots and cube roots of numbers less than 1. Had I digested and given all of that back in concise form this Instructable would have required considerably more steps. Over the years my most frequent uses of a slide rule have involved only a few squares or square roots and a very, very few cubes or cube roots. I think that could be true of many others, too. But, then my life's work does not involve a lot of mathematics. Still, I like to keep a slide rule next to where I sit while I am watching television. When a mathematical question comes into mind, I reach for that slide rule. Thank you for looking and for commenting.
I've never used one. I think I'll get one and learn.
If people know you are interested in slide rules, do not be surprised if someone comes to you with a slide rule that had belonged to someone in the family and the person who has it now would be very pleased to give it to you so it does not just gather dust. A nice Keuffel &amp; Esser rule from the late 1950s with all of the scales shown on the virtual slide rule in this Instructable can go for $30 to $50 US on eBay. I recently conducted a funeral for a man who had been an engineer. His widow knew from Facebook that I like slide rules. She gave me his K &amp; E Log-Log Trig slide rule. She was just happy it would be appreciated by someone. No one in the family had any interest in it. (I did find some corrosion had eaten the metal frames around the cursor and I spent about $15 US restoring it. The corrosion also left discoloration spots on the white celluloid skin of the slide rule. I am hoping sunlight will bleach that out. Anyway, it is fun to solve a problem with a slide rule while your friends are still looking around to see who has a calculator.
I just downloaded an iPod app called <em>Cube Slide Rule Lite</em>, just to use these instructions.<br>
Thank you. I think that is just about the highest praise anyone could give.

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Bio: I miss the days when magazines like Popular Mechanics had all sorts of DIY projects for making and repairing just about everything. I am enjoying ... More »
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