Intro: A More Complete Slide Rule Tutorial
A member at Instructables asked me to explain how to use a slide rule for the sake of those who came along about the time electronic calculators replaced slide rules. There are already a couple of tutorials posted at Instructables on using a slide rule. I aim to make this one more complete and easier to follow. It will cover all of the basic calculations, not just multiplication. Still, learning to use a slide rule is not a simple afternoon project. It takes time and practice over days, perhaps even weeks, for the principles to become part of the learner. You may want to download a PDF copy of this Instructable to your computer so you can learn the material in small bits.
You do not need your own slide rule because this Instructable will make use of a virtual Pickett N600-ES slide rule available for anyone to use at this link. It is the slide rule shown in the graphics. With the mouse cursor on your computer and your left mouse button, you can work the moving parts on this virtual slide rule just as if you were holding an actual slide rule in your hands. Just click and drag the moving parts at the link above. You can also drag the entire slide rule across your screen for a better view. Clicking and dragging will not work on the graphics in this Instructable. It is highly recommended that you open a second browser window and bring up the virtual slide rule so you can practice each new thing you learn in this Instructable.
The graphic shows the left end of the virtual slide rule with the various scales. From the top down you can see the LL1 or log-log 1 + and - scales; the A and B scales; the ST, T, and S scales; the C and D scales; the DI (D inverted) scale, and the K scale. Different makers of slide rules arranged the scales slightly differently according to what each maker thought would make its rules more user friendly. If you go to this virtual slide rule on the Internet, you will see a radio button on which you can click to display the scales on the back side of this virtual slide rule. Do not worry about that for now. Just be aware that this rule has two sides with additional scales on the backside. Because of all of its scales and features, a slide rule really is a full scientific calculator for all types of multiplication and division problems.
Some, but not all graphics in this Instructable do utilize text boxes. They are not always easy to see because the N600-ES slide rule used an eye saver (ES) yellow and black color scheme. Sweep your computer's cursor over key areas of the graphics to find any text boxes if you do not see them.
(Note: In my experience, some browsers may introduce some inaccuracies when using this virtual slide rule. If a reading is not quite accurate, try a different browser and see if that solves the problem. I will explain how to check for accuracy in a later step.)
Step 1: A First Check for Accuracy
Before using a slide rule new to you there is a preliminary check for accuracy that should be made. The inscribed line for the "1" on the A scale should be directly above the matching line on the B scale while the "1" on the C scale is directly above the line for the D scale. If this is not the case, the rule will need to be adjusted.
To adjust a slide rule, loosen the screwhead just to the left of the LL1 scale about a quarter turn. There is an identical screwhead at the right end of the rule. Loosen it a quarter turn, too. The top member of the slide rule's frame can now be moved a little to the left or right until the lines for "1" on all of the scales are directly inline with one another. Use a magnifying glass to be as accurate as possible. When all is in order, gently tighten the two screwheads again. The Pickett virtual rule shown is properly aligned.*
If you are using a plastic slide rule, it is permanently glued in place and (hopefully) is set accurately.
*Loosening these screws allows two adjustments. Already mentioned is aligning the top member of the frame so the 1 on the left side of each scale aligns with the 1 on the others. These screws also are used to adjust how much drag or tension there is on the movement of the slide rule parts when they are worked back and forth. If the slide rule is adjusted to be too loose, the center sliding portion can fall out and become damaged or lost. If the slide rule is adjusted to be too tight, using it for calculations becomes a lot of hard work.
Step 2: Basic Information About the C and D Scales
Locating a number on the scales of a slide rule is very much like reading any ruler. But, there are a couple of things that can be tricky and will need explanation.
The graphic shows the N600-ES virtual rule. It is a six inch rule. The smaller size allows it to be carried in a shirt pocket. Most common slide rules are ten inches in length. A longer rule is easier to read with accuracy because there is more space between the marks. (A PIckett N600-ES rule like this one went on the Apollo 11 mission to the moon and is now in the Smithsonian Institute in Washington, DC.) Despite the difficulty seeing yellow text boxes against the background of a yellow slide rule, I chose to use this rule for illustrating this Instructable because its shorter length made fitting images into the space available much easier.
For now ignore all scales on the slide rule other than the C and D scales. Notice that they are exact copies of one another. They are also the most often used scales, as well as the most basic.
A first thing to know is that the "1" at the right side (right index) of the C and D scales always represents a number that is 10 times larger than the "1" at the left side (left index) of the same scales. The left index could signify 10 and the right index would then signify 100, or it could be 1 and 10, or 1,000 and 10,000. The left index on the C scale could be 0.1 and the right index could be 1.0, while the left index of the D scale could at the same time be 10,000 and the right index could be 100,000. This all depends on the numbers in the problem to be solved. And, a slide rule gives only the significant numbers in the answer to any problem, not the placement of the decimal point. The user must determine that. There are a couple of ways to do that easily and they will be explained later.
For example, if you divide 540 by 3 on a slide rule, the slide rule will tell you that the significant numbers are 18. You will use some simple steps in your mind to know the answer is 180, not 1.8 or 18 or 1800, etc.
Step 3: Let's Read Some Numbers
The graphic has zoomed in on the left portion of the C and D scales. Notice the thin black line that crosses the graphic from top to bottom. This is called the hairline. It is used to mark and read numbers on the slide rule. It is inscribed on clear plastic or glass. The whole assembly is called the cursor. Some cursors have metal frames.
The hairline marks the significant digits 153. Find the longer mid-point mark between 1 and 2. This marks 15. Notice where 16 would be. There are four smaller marks between 15 and 16. These represent 152, 154, 156, and 158. The hairline falls directly over midway between 152 and 154, so it indicates 153. Examine the graphic until you can see clearly why the hairline marks 153.
If the left index is 1 and the right index is 10, this would be 1.53.
Step 4: Read Another Number
The graphic shows the C and D scales at the right end of the rule. The hairline marks 958, or perhaps 959. The difference between those two numbers amounts to a difference of only one-tenth of one percent, or the amount of accuracy that can be expected from a ten inch slide rule. Notice that there are fewer graduation marks at the right end of the slide rule, which makes accurately locating a number there more difficult. The longer mark between 9 and 1(0) indicates 950. You can easily find the mark that indicates 960. The hairline is positioned to represent 958.
Anyone who wants to use a slide rule will benefit greatly from lots of practice trying to locate numbers accurately. Randomly slide the cursor and read the number under the hairline where the cursor stops. I find I do my best reading of numbers when I do it in steps. I would look at this hairline setting and tell myself the first number is 9. I would notice the second number is greater than 5, but less than 6, so 5 is the second number. Then I would notice the third number is either 8 or 9 and probably 8. I would read the hairline as set at 958. Let's do some simple multiplication before I give any further detail related to establishing the number by means of where to place the decimal point.
Step 5: Multiplying 2 X 2 and Getting 4 As the Answer
Now it is time to do an actual multiplication problem on the virtual slide rule. You can replicate this by going to the link for the virtual slide rule in the Introduction and following the steps yourself. Move 1 on the C scale until it rests directly over 2 on the D scale. Move the cursor until the hairline comes to rest over 2 on the C scale. The answer will be on the D scale directly below the 2 on the C scale. The answer is 4. (In the past when I would teach someone how to use a slide rule, I described this as a lazy "Z" pattern. It is actually an inverted "Z" laid over on its side. You move downward from 1 on the C scale to the first factor on the D scale, then upward and across to the other factor on the C scale. Finally move downward to the answer. The series of movements forms an inverted "Z.")
Multiplication on a slide rule means adding the physical length associated with one factor to that of the second factor and reading the product directly below the second factor. More explanation of how and why this works will be offered later when logarithms are discussed.
Just for fun, look at the graphic again. Find 1.5 on the C scale. Notice 3 below it on the D scale. 2 x 1.5 = 3. Find 2.5 on the C scale. Notice the 5 below it on the D scale. 2 x 2.5 = 5. The slide rule can be very handy when you must multiply a series of numbers by the same factor. Simply move the hairline down the rule and the answers are already in place under each of the other factors. It is a little like entering one factor into the memory of an electronic calculator and pressing memory recall for each of the other factors in the series, except that there are fewer steps involved with the slide rule. This is an example of a problem that can be solved more quickly on the slide rule than on a calculator. With practice many people find they are as fast or faster on a slide rule than they are on a calculator.
Earlier I mentioned checking a slide rule for accuracy. I like to multiply 2 by a series of numbers, like 2, 2.5, 3, 3.5, 4, 4.5, and 5. The appropriate lines on the C and D scales should align exactly all of the way across, if the slide rule is accurate.
Step 6: Oops! What Do I Do Now?
This step will illustrate two things. First, it will show the multiplication of two numbers with far more digits than a simple 2 x 2 problem. The numbers I chose to multiply are 259 x 653. (This practice problem will reappear several times in the next steps to demonstrate alternative strategies for solving any problem. The purpose is to explain how additional scales can be used and how manipulation of the slide rule's parts can be done more efficiently.)
Set the 1 on the C scale over 259 on the D scale as you can see in the first graphic. But, when you locate 653 on the C scale, there is a difficulty. 653 on the C scale is suddenly hanging out in mid-air beyond the right index of the slide rule's D scale. See the second graphic for the solution to this difficulty. That solution is the second thing this step teaches.
Without moving the cursor or the hairline, move the sliding member of the slide rule (called "the slider") to the left so 1 on the C scale at the right index is directly under the hairline. Move your eye to the left and find 653 on the C scale. You may move the hairline on the cursor now for reading the product, even though I had to use a black arrow in the graphic to indicate the position of 653 on the C scale. I found the virtual slide rule a little difficult to read accurately on the D scale for this problem and my error rate was almost four tenths of one percent, which is quite high. When I used an actual ten inch slide rule, I had an accuracy error of less than one tenth of one percent. The numbers on the D scale of my actual slide rule were 169. If you do this problem on an electronic calculator, you learn the actual answer is 169,127. See the next step for judging the decimal point position.
Yet another way to do basic multiplication with a bit less moving of the slider will be mentioned in step 9. That means more efficiency of movement and more speed of use.
Step 7: That Decimal Point
Where to put the decimal point need not hang threateningly over our heads. A common and easy way to estimate where to place the decimal point is to do some radical rounding off of the factors. So, 259 x 653 would become 300 x 600 (or 700). 6 x 3 is 18. Add the four zeros for 180000. That is not far off from the actual answer of 169127 and tells you the answer will be in the hundreds of thousands, but less than millions and greater than tens of thousands.
Problems with smaller numbers are easier. Assume you wish to multiply 3.87 x 45.6. Round it off to 4 x 45 (or 50). 4 x 50 = 200. The actual answer from a calculator is 176.472. On a slide rule the answer you are able to read with certainty is 176. You can guess that the extra shown on the scale might add about 0.3 or 0.4.
Most people who are sufficiently interested in mathematics to use a slide rule are also very familiar with scientific notation. That means a large number can be written in a way that is much easier to handle. So, 1,000 becomes 1 x 103. That means the number is 1 with three zeros following it. 253 can be written as 2.53 x 102. The number 5 is 5 x 100, because anything to the zero power is always 1. The number 0.05 is 5 x 10-2.
So, (3 x 103) [or 3,000] x (5 x 10-1) [or 0.5] is a matter of simply multiplying 3 x 5 and adding the superscript numbers (exponents) together. A +3 and a -1 = +2. The answer would be 15 x 102, or 1500. Go back to the problem posed in step 6, which was 259 x 653. It could be written at (2.59 x 102) x (6.53 x 102). You know that 2 x 7 will be about 14. 102 x 102 means adding the exponent numbers and that indicates four zeros will be needed, so the answer is not far from 140000, at least in terms of where to put the decimal point.
Another way to handle the problem from step 6 is to simplify greatly one of the factors. So, 259 could become 2.59, but that means 653 must now also become 65300. The reason is that you divided the first number by 100, so you must multiply the second number by 100 to keep the problem on the same magnitude. Now it is fairly easy to multiply 2 or 3 by 65000 and know the answer is in the area of 130000 to 190000. Now the decimal places are clearly known.
If you have followed this Instructable this far, you have learned the basics of using a slide rule. Everything from this point forward is largely a variation on the process of multiplying with a slide rule.
Step 8: Division
Division on a slide rule involves the exact steps used in multiplication on the C and D scales, but in reverse order. Look at the graphic used earlier for 2 x 2 in step 5. To divide 4 by 2, find 4 on the D scale and place the hairline over it. Move the slider so 2 on the C scale is under the hairline. Find 1 on the C scale and the answer is directly under it on the D scale. It is 2.
Step 9: Division Another Way
Most slide rules include a CI scale (C inverted). It is identical to the C scale, but in reverse so that the numbers ascend from right to left, rather than from left to right. This virtual slide rule does have a CI scale, although you do not see it because it is on the reverse side of the slide rule. But, it does also have a DI scale, which is the reverse of the D scale. A CI scale and a DI scale are identical, except that the CI scale is on the slider and the DI scale is on the frame of the slide rule. You can see the DI scale on this rule just below the D scale. The "<" marks by the numbers indicate the scale is reversed. Also, the numbers, themselves, and the "<" marks are in red as a further indication that the scale is backwards. This is designed to reduce confusion when using the slide rule.
I have set the virtual slide rule in the graphic for the problem 6 divided by 2 = 3. I placed the 1 at the left index of the C scale over the 6 on the DI scale. See the black arrow. (Use the hairline to align 1 on C with 6 on DI.) Just as if I were multiplying, I moved along the C scale to find 2. The answer is under the hairline on the DI scale and is 3. Some people find this to be an easier way of dividing because it uses the same movement patterns used in multiplication.
The CI or DI scale may also be used for multiplication. See the second graphic. The advantage is that much less movement of the slider back and forth is necessary. To multiply 259 x 653 this way, locate 653 on the DI scale and place the hairline over it. (Remember that the DI scale progresses from right to left rather than from left to right, like the other scales.) Move the slider so 259 on the C scale is under the hairline. Read the digits 169 on the C scale above 1 on the DI scale. The manipulation of the slide rule parts is just like that involved in division using the C and D scales, except that one of the scales used is a reverse scale. This method of multiplication is actually a bit less confusing when using the CI scale located on the slider (backside of the virtual slide rule) rather than the DI scale located on the rule's frame (on the front side of the virtual slide rule).
Step 10: Proportions
If five widgets fit into three quarks, how many widgets would fit into 13 quarks? That is a proportion problem. 5 is to 3 as x is to 13. It can be written as 5/3 = x/13, or it can be written as 5:3 : : x:13. When written with colons as separator devices it is easier to understand the rule that in a proportion the product of the means equals the product of the extremes. The extremes in this equation are 5 and 13. the means are 3 and x. So, 3x = 5 x 13. See the second graphic. Set the hairline at 13 on the D scale. Move the slider so 5 on the CI scale appears under the hairline. This multiplies 5 by 13. Find 3 on the CI scale. Move the hairline over it. Read 21.6 as the answer on the D scale.
The Dietzgen manual that is linked in the final step of this Instructable offers this way to solve a proportion problem. Imagine the proportion x:75 : : 88:60. See the third graphic. Move the hairline over 88 on the D scale. Move the slider so 60 on the C scale appears under the hairline. Without moving the slider, move the hairline to 75 on the CF scale. Read 110 under the hairline as the answer on the DF scale. (Decimal Trig Log Log Type Slide Rule by Ovid W. Eshbach, published by Eugene Dietzgen Company, 1956, p. 27. The CF and DF scales will be discussed in step 14.)
Step 11: Reciprocals
Sometimes you need to know the reciprocal of something. That means you want to know what 1 divided by a number is. The CI scale makes it easy. Just find the number on the C scale with the hairline and read its reciprocal directly above on the CI scale. Or, you could use the D and DI scales. If you need to know the reciprocal of another number, just move the hairline and read the answer.
In the graphic the pale blue letters mark the C and CI scales. The hairline is over 2 on the C scale. The reciprocal of 2 is 0.5, as you can see on the CI scale under the hairline. Without moving the slider, you can also see by the blue arrows that 0.4 is the reciprocal of 2.5, and by the red arrows that 0.333 is the reciprocal of 3.
Step 12: Squares and Square Roots -- the a and B Scales
The A or B scale is used in conjunction with the C or D scale for quickly finding the square or square root of any number. If you look closely at the A and B scales, they are replicas of the C and D scales, but made smaller so two complete scales fit end to end on the A or B scales in the same space given to the C or D scales.
Move the hairline over a number on the D scale and read its square on the A scale (or use the C scale and the B scale). So, the hairline is over 2 on the D scale. Move your eye upward and you can see the number under the hairline on the A scale is 4. 2 x 2 or 2 squared = 4. This problem could also be 20 squared, in which case, the answer would be 400. The user is responsible for placing the decimal point. I wanted to use only one graphic for this step, so I made black arrows in place of the hairline to show that 8 squared is 64.
Squares are easy. Square roots are a slight bit more complicated. The square root of numbers with an odd number of digits are solved on the left end of the A or B scale. Numbers with an even number of digits are solved on the right side of the A or B scale. I often forget which is which, so I do a quick check by squaring first 2 and then 8 to see on which end of the A scale the answer for each is found. Squaring 2 results in a product with one digit, an odd number. Squaring 8 results in a product with two digits, an even number.
The process is a little more complicated when solving the square root of numbers less than 1. So, the square root of 0.4 is 0.632 (right half of the A scale), but the square root of 0.04 is 0.2 (left half of the A scale). For more on those rules, see the manual linked in the last step of this Instructable. Otherwise, this Instructable will become an entire book rather than a few steps.
Bonus: Back in step 6 we ran into the problem of one factor hanging out in midair and then the user needed to move the slider back nearly full length to read the answer. It is possible to use the A and B scales for multiplication. If the slider moves beyond the range of one side of the A or B scale, you can read the answer on the other half of the A or B scale just fine. The only drawback is that the A and B scales are smaller and more compressed, which makes accuracy in finding and reading numbers more difficult. See the second graphic. The familiar practice problem of multiplying 259 by 653 has been set up on the A and B scales. 1 on the B scale is aligned with 259 on the left half of the A scale. Move the hairline to 653 on the left half of the B scale. The answer 169000 can be read on the right half of the A scale.
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.
Step 14: Folded Scales
The first graphic shows the reverse side of the virtual slide rule. The C and D scales are where you have come to expect them from the other side of the slide rule. Notice that where the A and B scales were on the front side of the slide rule you see a CF and a DF scale. These two scales are identical to the C and D scales, except that they are offset by the amount of pi. At the left index you do not see 1, but you see the setting for pi. (Sorry, but I could not get the Greek letter to print in this text. Even the special character command did not work.) If you want to know what any number multiplied by pi is, simply find the number on the C or D scale. Move the hairline over this number and follow the hairline up to the CF or the DF scale to read the answer.
In the graphic the hairline is over 15 on the D scale. 15 x 3.1416 or pi is 47.2 on the DF scale. (The actual answer worked on an electronic calculator is 47.12, which is quite close.) From this point, more steps in a problem involving pi can be finished on the CF and DF scales by multiplying just as if they were the C and D scales. The advantage is that there is no need to transfer the product of pi and another number down to the C and D in order to finish the problem.
Here is a sample problem begun on the D scale and finished on the DF scale. (Use the button on the virtual slide rule page to turn the slide rule to its back side.) See the second graphic. You wish to know the circumference of a circle with a 3 inch radius. The formula is Circumference = pi x radius x 2 (or pi2r). I have included pale blue letters to mark the D, DF, and CF scales. Move the hairline over 3 on the D scale. Move your eye up to the DF scale. The number under the hairline equals pi x 3, but there is no need to read it because the problem is not yet finished. Move the slider so 1 on the CF scale is under the hairline. Move your eye to the right to locate 2 on the CF scale. The solution to your problem is above the 2 on the DF scale. The circumference of a circle with a radius of 3 inches appears to be a bit more than 18.8 inches. According to an electronic calculator, the correct answer is 18.85 inches.
Another advantage to the CF and DF scales is that they have the effect of making the slide rule longer because they are offset. You may remember back in step 6 the second factor in a multiplication problem at first hung out in the air beyond the right end of the rule. Because problems may be finished on the CF or the DF scales, the difficulty experienced in step 6 is much less likely to happen.
You may already have noticed the location of pi is also marked with the Greek letter on C and D scales as well as on the A and B scales. No matter which of these scales you are using for a problem, you can find pi quickly and easily.
One additional piece of useful information--The N600-ES is a duplex slide rule. That means if the hairline is over 2 on the D scale on the front side of the rule, you can turn the rule over and the hairline on the backside of the rule will also be over 2 on the D scale. Some problems require the use of scales on both sides of the slide rule. These problems can be worked by simply turning the rule over and working the next part of the problem. Turn it back to the other side again as needed. No inaccuracies will enter your calculations. Nothing will be lost. There is no need to record a number on paper and then locate it on a scale on the other side of the rule.
Step 15: Trigonometry Functions
Trigonometry allows you to solve triangles. It requires a right triangle, that is, a triangle with one 90 degree corner. If you are solving a triangle with no 90 degree corner, you can bisect one of the angles and extend the line so it intersects the opposite side at 90 degrees. See the first graphic. Then you solve either or both of those triangles to find the solution to your problem. Notice the green line and how it forms two right triangles from a triangle in which there were no right angles.
I once wanted to know how tall a tree was without climbing it. I used basic trigonometry to solve my problem. See the second graphic.
Triangle sides have names. The long sloping side is the hypotenuse. The side next to the angle that is known (other than the 90 degree corner) is the adjacent side. The side that defines the height of the tree in the graphic is the opposite side.
A teacher once summarized basic trigonometric functions with the word soh-cah-toa, which means Sine = Opposite divided by the Hypotenuse, Cosine = Adjacent divided by the Hypotenuse, and Tangent = Opposite divided by the Adjacent. In my problem concerning the height of the tree, I knew the angle between the Hypotenuse and the Adjacent sides, and I knew the length of the Adjacent side. I wanted to know the length of the Opposite side. That is a problem involving a Tangent. Factoring, the equation became: Opposite = Tangent x Adjacent. I was about 40 feet from the center of the tree's base. The angle indicated by the red arc was about 61 degrees. I determined the tree was 72 feet tall. The Tangent of 61 degrees is 1.804. So, 40 feet x 1.804 = 72 feet.
See the third graphic. If you need to know the exact value of any trigonometric function (tangent, cotangent, sine, cosine, secant, cosecant), open an Internet search engine, like Google or Bing, and enter this string customized to your purposes: tangent 61 degrees. Change the function and the angle to fit your needs. The number to multiple decimal places will appear at the very top of the returns.
If you are in a hurry and want to solve a triangle without attempting the process on a slide rule, go to this link. If you want to learn a bit about using a slide rule to solve problems involving trigonometry, continue to the next step.
Someone will surely say, "Why not just step back until you can look up at 45 degrees and see the top of the tree? Then all you need to do is to measure the distance to the base of the tree and you know that is equal to the height of the tree." While that is true, a neighbor's fence could be in the way, or a body of water. There are times when you just need to solve problems with trigonometry functions.
Step 16: Calculate the Height of the Tree on the Slide Rule
In the previous step, I viewed the top of a tree through a camera with a telephoto lens tilted to 61 degrees. I did this from a distance of 40 feet. By means of the tangent of 61 degrees, the tree was determined to be 72 feet tall. Follow these steps to see how that is done on the slide rule.
Place the hairline over 40 on the D scale. Notice the pale blue letter in the graphic to indicate the D scale and also the T scale a couple of scales above it. The T scale is marked in degrees of angle. But, see the red circle at the right index of the T scale. It goes only to 45 degrees when moving from left to right. Notice also the red numbers on the T scale with the red "<" marks. Angles above 45 degrees are read moving from right to left and using the red numbers on the T scale. That sounds confusing, but here is the secret. Angles at 45 degrees or less (black numbers) are read on the C scale (or the D scale if both are aligned). Angles greater than 45 degrees are read on the CI (or DI scale if the C and D scales are aligned). Also, the tangents of angles less than 45 degrees are less than 1 (decimal numbers). The tangents of angles greater than 45 degrees are greater than 1. Keep this in mind if you need to know where to place a decimal point. Next, recall from the previous step that the tangent of 61 degrees was found to be 1.8. In the graphic 61 degrees on the T scale has been placed under the hairline. Take a moment to see why the hairline reads 61 degrees on the T scale. Now, recall what was done when the reverse CI or DI scale was used with the C or D scales for multiplication. The red numbers on the T scale are also a reverse scale. The second factor is placed over the first when a reverse scale is used, just as if you were dividing on the C and D scales; but, you are actually multiplying because one of the scales is a reverse scale. Placing two factors one above the other when one of the scales is a reverse scale multiplies, not divides. Then move your eye to the 1 at the right index of the C scale. Read the answer below it on the D scale. The tree is a tiny bit more than 72 feet tall, as indicated by the red arrow. There is no need to read the actual tangent of 61 degrees and know that it is 1.8. All you need to do is to locate 61 degrees on the T scale and move the slider as in any other multiplication problem involving a reverse scale so you can read the answer under the 1 at the right or left index of the C scale.
Step 17: Cotangents
The slide rule has scales for Tangents and for Sines. But, there are also Cotangents, Cosines, Secants, and Cosecants. The scales for Sines and Tangents do allow the user to find and use Cotangents and Cosines as well as Sines and Tangents. If you know the relationship of the various trigonometric functions, you can also calculate Secants and Cosecants.
The previous step explained how to find Tangents for angles up to 45 degrees by using the T and the C scales, and for angles greater than 45 degrees by using the CI scale. But, if angles up to 45 degrees on the T scale are read on the CI scale instead of the C scale the result is actually the Cotangent. Likewise, if angles greater than 45 degrees on the T scale are read on the C scale rather than on the CI scale, the result is the Cotangents for those angles. So, for example, the Cotangent for 60 degrees is the same as the Tangent for 30 degrees, and vice versa. Unless you work with triangles a lot, you may not even wish to bother about this.
The photo shows the T, ST, and S scales on the Dietzgen #1733 slide rule I bought on eBay. This rule uses red numerals for reverse scales, but does not include "<" marks.
Step 18: Sines and Cosines
I wish my problem in which I needed to determine the height of the tree had involved Sines because Sines are easier to determine on the slide rule than are Tangents and Cotangents. It would have been an easier, more logical progression to explain Sines and Cosines before explaining Tangents and Cotangents.
The S scale runs from left to right and goes up all of the way to 90 degrees. It is marked in angles. Find the angle and read or mark its Sine on the C scale. Notice the S scale also has red degree numbers running from right to left. Use the red numbers when you need to know or use a Cosine. They are also read on the C scale. That means the Sine of 8 degrees is the same value as the Cosine of 82 degrees.
Many, but not all trig(onometry) slide rules have an ST scale. This scale is used for calculations involving very small angles of about 5 degrees or less. See the manual linked in the last step on how to use this scale.
Step 19: Logarithms
Exponents are logarithms. Exponents for base 10 were discussed in step 7 as part of scientific notation and knowing where to locate the decimal point in solving a problem.
Every number has a logarithm number. Turn the virtual slide rule over to its backside. Notice the L scale on the slider. Logarithms are why a slide rule works. Move the hairline over 2 on the C scale. Look at the L scale. The logarithm of 2 is 0.303. Add 0.303 to 0.303 and the answer is 0.606. See the second graphic. Move the hairline to 0.606 on the L scale. Look down to the C scale. Notice that 4 is under the hairline on the C scale. Adding the logarithm of 2 to the logarithm of 2 gives the logarithm of 4. Adding logarithms is the same as multiplying the numbers represented by the logarithms (antilogarithms). The slide rule converts numbers to lengths that represent the logarithms of those numbers and adds them or subtracts them to multiply and divide.
Logarithms are a simpler way to work with very large numbers by adding and subtracting their exponents. The digits to the right of the decimal point are the mantissa. Numbers to the left of the decimal point are called the characteristic. It indicates how many zeros are part of the number, as in where to put the decimal point in the solution to the problem. The logarithms for the L scale are base 10.
The LL1, LL2, and LL3 scales, or log-log scales, are used for numbers in bases other than 10. For example, hexidecimal calculations involve numbers in base 6. The LL scales are frequently used for various types of engineering problems and can even handle factors with different bases at the same time. Unless you are an engineer and know formulae that require log-log calculations, you will not have much need for the LL scales.The manual linked in the final step does have several pages that make a nice, easy to understand introduction to logarithms and all of the logarithm scales, including the LL scales, except the Ln scale (natural logarithms) on the virtual slide rule. You can search on-line to find a manual for the PIckett N600-ES slide rule and it will explain the use of the Ln scale. By the way, that scale was invented by a high school student in the 11th grade.
The LL scales make easier work of calculations with numbers between 1.001 and 20,0000. Basically, a number is located on the appropriate LL scale according to the size of the number and its logarithm is read on the C or D scale. The logarithms are added or subtracted. The resulting logarithm is located on the appropriate scale and converted back to a number for the solution to the problem. Log-log problems could be solved in a more conventional way, but that would require much more work to arrive at the solution. LL scales save time and effort. They make big problems easier (for the person who thoroughly understands how to use them).
Step 20: Convert Temperature Readings
By means of Facebook and e-mail contacts I often need to covert temperatures in Fahrenheit to Celsius, and vice versa. You can just type "25 degrees Celsius to Fahrenheit," etc. in the search window of your favorite search engine and the answer will appear at the top of the hit list, but you could have only a slide rule nearby. Pickett once made a slide rule with metric conversion scales on the backside of the rule. But, this step is a little bonus for the person who will use a slide rule regularly, but the slide rule does not have those special conversion features.
To convert from Fahrenheit to Celsius, subtract 32 from the temperature reading in Fahrenheit and divide by 1.8. Slide rule manipulation-- After subtracting 32 from the temperature in Fahrenheit, locate the remainder on the D scale with the hairline. Move the slider so 1 on the CI scale appears under the hairline. Move the hairline to 1.8 on the CI scale. Read the answer under the hairline on the D scale. In the first graphic the CI, C, and D scales are marked with the pale blue letters at the right side of the graphic. The first graphic illustrates converting 82 degrees Fahrenheit to its equivalent in Celsius. 82 - 32 = 50. 1 (right index) on the C scale has been placed over 50 on the D scale. The hairline is over 1.8 on the CI scale. Read the answer on the D scale. It is 27.7 degrees Celsius.
To convert from Celsius to Fahrenheit, multiply the temperature in Celsius by 1.8, and add 32. Slide rule manipulation---Move the hairline to the temperature in Celsius on the D scale. Move 1.8 on the CI scale so it appears under the hairline. Read the number under the right or left C scale index on the D scale. Add 32 to this number. The second graphic illustrates converting a temperature in Celsius to its equivalent in Fahrenheit. The left index on the C scale has been placed over 20 on the D scale. Follow the C scale to the right on the C scale to the hairline. It is over 1.8. Read 36 below it on the D scale. Add 32 to 36 for a total of 68. The equivalent of 20 degrees Celsius is 68 degrees Fahrenheit.
Step 21: Conclusion
Most who will want to learn to use a slide rule now will do it out of nostalgia. Their father may have used a slide rule during his engineering career and they wish to honor their dad by learning to do calculations as he did them. Perhaps they have inherited his very nice vintage slide rule. Or, like the Instructables member who requested this Instructable, they came along just after electronic calculators had replaced slide rules, and they would like to learn how slide rules were used. Some will be enamored of the possibilities of learning something retro, as they were living in the old days, themselves, just like those who enjoy historical re-enactments.
Before you decide there is absolutely no point in giving thought to a slide rule, consider this. A friend served in the US Army Reserves as a weekend warrior. He was in artillery. The field gun pieces they fired in their exercises were all computer controlled. Enter some data and the computer gave all of the information needed for aiming the guns. One weekend the computer system went down and would not work. The older guys in the group had learned gunnery before the days of computers and one of them had brought the slide rule calculator and the data charts to go with it. His team was able to aim and fire their guns, but the other guys who were dependent on the computer could not. Imagine if this had been a battle during a war, and not just a field exercise. Knowing how to do the manual calculations on a slide rule is a desirable skill to have, just in case there is a problem one day.
A manual for a good vintage slide rule about 100 pages. There is no way I can give all of that detail in an Instructable that has 21 steps plus an Introduction. I learned slide rule basics by myself with the much shorter 3-step manual for the Pickett 120 Trainer plastic slide rule. Scroll down to or search the linked page for "M52." I have also been working my way through a Dietzgen manual, and it is very well done. You can download a free copy here. Scroll down to or search the page for "M159 ref S454" for a log log slide rule manual. There are plenty of practice exercises with an answer table at the back of the manual. The Dietzgen manual is also very good about mentioning little tips and tricks to make using a slide rule faster and easier. I have tried to include some of those as much as is practical to do within the limitations of an Instructable.
If, after practicing on a virtual slide rule, you decide you want your own slide rule, there are several sites on the Internet that sell them. Some are pristine and are sold for collectors at premium prices. Many slide rules of all types and conditions are sold in North America on eBay. Slide rules sold there sometimes have a reputation for not being as described or for having broken or missing pieces. See my previous Instructable on restoring a vintage Dietzgen slide rule I bought at eBay for a very fair price. And, I once did an Instructable on making your own slide rule. Both have some very useful links to other slide rule sites on the Internet. If you would like to make your own very nice circular slide rule, see this Instructable. There are some charges for engraving. You can buy a new plastic slide rule that appears to be very much like the Pickett 120 Trainer I used in high school. After reading reviews on this Think Geek rule, I would look for a Pickett 120 Trainer or 140 on eBay as my first choice. The 120 does not have as many scales as the 140. When electronic calculators replaced slide rules Pickett had a warehouse full of plastic rules that were bought up and are still being sold as new, never used rules. These are good basic slide rules that will serve well, although the 120 does not have all of the scales you see on the virtual slide rule used in this Instructable. My Pickett 120 still works very well after 50 years. (One of the most famous slide rules is the Nestler 23, a German made rule with the same few basic scales found on my Pickett 120 Trainer. Yet, it is the slide rule used by both Sergei Korolev, head of the Russian space program, and Werner von Braun, head of the USA space program.)
Some will ask if it is possible to add and subtract using a slide rule. There have been people who worked out ways of doing that. From what I have read those methods are innovative, novel, complicated, limited, and not very practical. Slide rules are really designed for various calculations involving multiplication and division.
Unless you are an engineer, you will not use all of the features built into a slide rule. But, a slide rule does force the user to be more involved in any mathematical calculation. If you like mathematics, you will enjoy learning and using a slide rule, and you will gain a certain sense of personal satisfaction from mastering it.
The photo shows the Pickett 120 Trainer I bought 50 years ago in 1961. The case and the manual show some wear. The slider once fell out while I was waiting for a school bus. A day later when I found it, there were some surface blemishes from gravel and dirt on the driveway after a car had driven over it.