Published on the Doomstead Diner on April 17, 2016
Discuss this article at the Science & Technology Table inside the Diner
One of the many things we have come to take for granted in the Age of Oil is the ease with which we can do long and tedious calculations, these days with electronic computers, but even before that with electronic calculators,, and before that with various types of mechanical adding machines. Nowadays, the ubiquity of machines which will do all your daily calculations has resulted in a population largely unable to just do basic arithmetic of adding, subtracting, multiplying and dividing without the aid of a calculator.
In general, if you can at least do those things, this covers you most daily math tasks a person will face in a day, unless you start building things in which case you start having to deal with angles and geometry and (gasp) even trigonometry, and then it gets worse still if you have to start multiplying up or dividing fairly large numbers, like say you have 173 Acres of land with each acre averaging 19,400 lbs/acre of organically grown carrots and you have 97 people living on your SUN☼ community property, how many pounds of carrots each year will each SUN☼ community member get?
ORGANIC CROP INFORMATION FROM:
The Owner-Built Homestead, by Ken &Barbara Kern
|#||VEGETABLE NAME||YIELD: POUNDS / ACRE|
I can tell you quite easily how many poiunds each person gets, it is 173X19400/97=34,600 lbs of carrots! That is a LOT of carrots, more than I could eat in a year anyhow. Of course, your whole 173 Acres will not be dedicated to monoculture growing of carrots, so there will be a good deal less than that amount of carrots each year coming off the property. This is just an example of a typical math problem you might face though in terms of allocating how much land to grow carrots on, how much for potatoes, how much for spinach…etc.
So how did I get that result? Well, I'm an ex-math teacher so I can ballpark this kind of calculation in my head, but I didn't do that in this case, I just Googled up a Calculator, plugged in the numbers, hit ENTER, and POOF I got the correct result! No muss, no fuss, and really fast too!
What if I did NOT have an electronic calculator at my fingertips though? Well, I could do the multiplication and long division on paper, but this is time consuming and tedious shit. Do you think Einstein did all his calculations on paper this way when he was checking out his Special Theory of Relativity? OF COURSE NOT! Einstein would still be doing the calculations today if he had tried to do it that way.
Einstein and just about every scientist or mathematician before 1970 or so who wanted to check the validity of his equations by plugging in some numbers did so by using a Slide Rule. Slide rules traditionally were manufactured in a linear format, like a regular ruler. Except you didn't use them to measure the length of lines, you used them to do quick calculations on large numbers, and to get numbers you would otherwise need to look up in a table, like logarithms and trigonometric functions.
How accurate you could be with one of these things depended on several factors:
1- How big the slide rule was. The bigger the better to spread out the scales.
2- How well machined the slide rule was. Scales had to be accurately scribed onto the rule, and the sliding part had to be very solid and not wobbly.
3- How good your eyesight and ability to estimate fractions of a space were.
Back in those days, a really high quality slide rule (usually from a German manufacturer) could cost a lot of money. For myself in HS, I just had a cheap 6" job that served its purpose pretty well. I still have it too! 🙂 As I headed off to college though, the first of the scientific calculators were being produced by Texas Instruments and Hewlett-Packard, and I coveted the HP-45, the most expensive of these coming in at $450 in 1970s dollars. I worked all summer in a shower curtain factory to afford this fabulous gizmo. With it, you could calculate everything a slide rule could, and with far more accuracy to around 12 digits as I recall. It was more expensive than the finely machined German slide rules, but actually by not that much and way more accurate and EZ to use.
These devices also got cheaper quite quickly, and by the time I graduated were about 1/2 my purchase price. Now you can get similar models for $20 of 2010 dollars. So they put the Slide Rule manufacturers out of bizness pretty quickly, except a few who produce them as novelty items and nostalgia items for aging scientists. lol.
Since they are no longer made or used much anymore, nobody learns how to use them either. Not even in science and math prep schols like Bronx High School of Science or my alma mater, Stuyvesant HS. It's one of those lost arts that probably fewer people know how to do than know how to make a stone axe out of obsidian. lol.
Recently however I began thinking about how we will do calculations for building things in a post-industrial world, stuff which we still have materials to build with and a reason for building the device. Wooden bridges to span medium size rivers, trebuchets to hurl boulders at the enemy, that sort of thing. LOL. No electronics in this projected world of the future, so we need to return to the Old Ways in this area also.
We're also not likely to have the fine machining capabilities of German factories so making really good linear slide rules would be almost as difficult as making an HP-45. Is there an answer to this problem? Yes, there is. ROUND SLIDE RULES! You can make these out of paper or cardboard or even wood or stone as long as you can cut out an accurate circle with accurately placed center, which is not that hard to do. Having Accurate scales to scribe onto your circle is necessary also, you can make these from scratch but it is tedious. Fortunately, there are numerous Circular Slide Rule templates available for FREE download online, and you can print out few to have a hard copy available when the grid goes down for good.
A serviceable slide rule is not the only thing you'll need around to to good engineering in the post collapse world of course, to make your circles to begin with you'll want to have a good Compass as well. They're not electronic so you can make one of them if need be out of a couple of sticks, but you probably won't be able to tune the size of the radius as accurately as a well machined one with a screw adjustment. For doing your designs at scale, one of these will be very handy to have in your preps.
When you are ready to scale up to your full size Trebuchet capable of hurling tons of rock at the Zombies or you want to build a Sun Dial or a Solstice calculator like Stonehenge on your SUN☼ property, you'll need to be able to measure out and draw much bigger circles. How do you do that?
Fortunately that is actually easier than the small circles for the design, all you need is some rope at the desired length of the radius and a stake driven into the ground at the center of that radius. Then walk around the circle holding the rope or string tight, and your footsteps will trace out a perfect circle. Your big circle will also prove valuable for scaling up anything, since you can use fractions of it's total radius and get a good measurement. You'll also want to have Angles marked off on both your large and small circles to do angle measurements, important for any engineering design. Marked off into 360 degrees, these circles are known to most grade school children as Protractors. A good protractor better than a cheap plastic one is another good thing to keep in your preps if you want to make accurate measurements in the post-collapse world.
However, what if you neglected to keep such a valuable tool in your bugout bag? Can you make one of reasonable accuracy with just your straight edge and compass or string & stake arrangement? Yes you can!
Dividing your circle into halves of 180 degrees is EZ, just use your straight edge to draw a straight line from the edge of the circle throuh the center to the other side. Now you have 180 degree markings on your circle.
Step 2, Bisect the line forming the diameter of your circle with your compass. I won't explain that, you should have learned how to bisect a line with a compass by the 6th grade. Draw a straight line through the bisection points, and now you are down to 90 degrees. Connect those points on the circle and you now have a perfect square. Bisect each of those lines, connect to the center, and now you are down to 45 degrees, the 8 main direction points of a Navigation compass, N, NE, E, etc.
Getting 30 degree increments is a little trickier, for this you need to take your compass set at the original radius of the circle, put the center on the edge of the circle and then draw another circle. Then you move around the circle placing the point of the compass on each place the circles intersect and draw a new circle, and then you keep moving around the original circle until you return to your start point. Assuming you did good drawing, you will get exactly 6 circles surrounding the first circle. This is called "close packing", and it's the reason Honeycombs for Bees are shaped like Hexagons, Snowflakes have 6 points and numerous other examples in the natural world.
Connect up the intersection points to each other, and you will get a perfect regular hexagon, connect to the center and you will now have 60 degree angles. You also already have 45 degree angles, so breaking up into 15 degree increments is EZ, just use your compass on the radius to measure the distance on the circle edge between the 60 and 45 degree marks and work your way around the circle marking them off as you go. You will get from this 24 increments of 15 degrees each, which is where your 24 hour clock and 12 hour analog clock face come from.
Your final divisions to get down to 360 are a bit trickier than getting down to 15 degrees. 15 is the product of two Prime Numbers, 5 & 3. So in order to get down to perfect 1 degree increments so you can be nice and accurate with your engineering projects, you have to divide this by 5 and by 3, with nothing but your compass and straight edge.
Trying to figure out how to do this is pretty tough, but fortunately a bunch of Greeks with nothing better to do with their time like Pythagoros and Archimedes worked out some methods for this a couple of thousand years ago. In the effort to not let this bit of knowledge be forgotten as we move down the collapse highway, I am disseminating it here.
To divide into 5ths, you'll need to divide your circle up into a Regular Pentagon. The interior angles of a regular pentagon are all 72 degrees. If you can do this, then 72 degrees minus 60 degrees gives you a 12 degree angle, and then a 15 degree angle minus a 12 degree angle gives you a 3 degree angle. 🙂
The problem of course here is dividing up the circle into a regular pentagon, not so EZ as a regular hexagon. Explaining how to do this in text without illustration is just about impossible, however fortunately there are Utoob tutorials on how to do it. Here is one:
The final 3 degree split to get you down to 360 one degree increments is really the toughest bear, because there is no really "pure" way to trisect an angle. When I make an impromptu protractor, I just fudge this part and eyeball it. This is good enough for most purposes, in fact for most engineering type purposes you will rarely find you need anything below 15 degrees, and 3 degrees is enough for even geodesic domes, which have pretty complex angles in them.
However, Archimedes and a few others who also had way too much time on their hands like I do did work out methods for trisecting an angle using fudges of their own. Here is the full description of how to trisect an angle from the UIUC Geometry Forum:
Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a circle among other constructions.
A few weeks ago I explained my job to a group of professors visiting the Geometry Center . I mentioned that I wrote articles on a newsgroup about geometry and that sometimes people write to me with geometry questions. For instance one person wrote asking whether it was possible to divide a line segment into any ratio, and also whether it was possible to trisect an angle. In response to the first question I explained how to find two-thirds of a line segment. I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle. One professor I told this story to replied by saying, "Bob it is possible to trisect an angle." Before I was able to respond to this shocking statement he added, "You just needed to use a MARKED straightedge and a compass." The professor was referring to Archimedes' construction for trisecting an angle with a marked straightedge and compass.
When someone mentions angle trisection I immediately think of trying to trisect an angle via a compass and straightedge. Because this is impossible I rule out any serious discussion of the manner. Maybe I'm the only one with this flaw in thinking, but I believe many mathematicians make this same serious mistake.
Why tell people it is impossible to trisect an angle via straightedge and compass? Instead we could say it is possible to trisect an angle, just not with a straightedge and a compass. When told that it is impossible to trisect an angle with a straightedge and compass people then often believe it is impossible to trisect an angle. I think this is a mistake and to rectify my previous error I will now give two methods for trisecting an angle. For both methods pictures are included that will hopefully illuminate the construction.
The first method, Archimedes' trisection of an angle using a marked straightedge has been described on the Geometry Forum before by John Conway. First take the angle to be trisected, angle ABC, and construct a line parallel to BC at point A.
Next use the compass to create a circle of radius AB centered at A.
Now comes the part where the marked straightedge is used. Mark on the straightedge the length between A and B. Take the straightedge and line it up so that one edge is fixed at the point B. Let D be the point of intersection between the line from A parallel to BC. Let E be the point on the newly named line BD that intersects with the circle. Move the marked straightedge until the line BD satisfies the condition AB = ED, that is adjust the marked straightedge until point E and point D coincide with the marks made on the straightedge.
Now that BD is found, the angle is trisected, that is 1/3*ANGLE ABC = ANGLE DBE. To see this is true let angle DBC = a. First of all since AD and BC are parallel, angle ADB = angle DBC = a. Since AE = DE, angle EAD = a, and so angle AED = Pi-2a. So angle AEB = 2a, and since AB = AE, angle ABE = 2a. Since angle ABE + angle DBC = angle ABC, and angle ABE = 2a, angle DBC = a. Thus angle ABC is trisected.
The next method does not use a marked ruler, but instead uses a curve called the Quadratrix of Hippias. This method not only allows one to trisect an angle, but enables one to partition an angle into any fraction desired by use of a special curve called the Quadratrix of Hippias. This curve can be made using a computer or graphing calculator and the idea for its construction is clever. Let A be an angle varying from 0 to Pi/2 and y=2*A/Pi. For instance when A = Pi/2, y=1, and when A=0, y=0. Plot the horizontal line y = 2*A/Pi and the angle A on the same graph. Then we will get an intersection point for each value of A from 0 to Pi/2.
This collection of intersection points is our curve, the Quadratrix of Hippias. We will now trisect the angle AOB. First find the point where the line AO intersects with the Quadratrix. The vertical coordinate of this point is our y value. Now compute y/3 (via a compass and straightedge construction if desired). Next draw a horizontal line of height y/3 on our graph, which gives us the point C. Drawing a line from C to O gives us the angle COB, an angle one third the size of angle AOB.
As I mentioned before this curve can be computed and plotted via a computer. The formula to find points on the curve is defined as x = y*cot(Pi*y/2). Yes here the vertical variable, y, is the independent variable, and the horizontal variable, x, is the dependent variable. So once the table of values is found, the coordinates will need to be flipped to correctly plot the Quadratrix.
Now a justification for the formula. In the following figure, B =(x,y) is a point on the Quadratrix of Hippias. Let BO be a line segment from the origin to B and and BOC be our angle A. If we draw in a unit circle, and drop a vertical line from the intersection of the angle we get similar triangles and see that sin(A)/cos(A) = y/x, or tan(A) = y/x. But earlier we defined a point (x,y) on the Quadratrix to satisfy y=2*A/Pi. So we get tan(Pi*y/2)=y/x or equivalently x = y*cot(Pi*y/2).
So we now have two different ways of trisecting an angle. I learned about the construction of the second method in Underwood Dudley's book: "A Budget of Trisections". In the book Dudley describes several other legitimate methods for trisecting an angle as well as compass and straightedge constructions that people have claimed trisect an angle. The book also contains entertaining excerpts of letters from these "angle trisectors".
Besides stating it is impossible to trisect an angle, I think other problems occur in discussing angle trisection. One difficulty is in explaining what it means for something to be impossible in a mathematical sense. I definitely remember in high school being told that it was impossible to trisect an angle. But I think at the time it meant the same thing to me as that it was impossible for me to drive a car. I was only 14 years old and I could not get a license to drive a car for another two years so it was just not possible AT THAT TIME. I do not remember being told that when something is impossible in mathematics, it was not possible five million years ago, it is not possible now, and it will never be possible in the future. Granted I may not have been ready for an explanation of mathematical logic and proof, but a statement like, "It is impossible to trisect an angle with a straightedge and a compass. This means it is no more possible to trisect an angle with those tools, then it is to add 1 and 2 and get 4" would have been much more powerful. (I am assuming here that we all count the same way: 1, 2, 3, 4, …)
I did not intend to attack my high school teacher; I learned an incredible amount of mathematics from her as well as a deep love for the subject. Maybe my teacher did explain what the words "mathematically impossible" meant, and I just do not remember her comments. Regardless, I think a discussion of impossible in the mathematical sense would be an interesting and valuable topic to discover in high school. Are there any teachers out there who have spent time talking about mathematical impossibilities?
I was going to detail methods for creating your own rulers for measuring distances and methods for creating your own measures of mass and weight and all the rest of the stuff we deal with in the 3D World, but I think the class is mostly asleep by now so I will stop for today for recess. Go out and PLAY!