I'm taking a little bit of time away from writing about the culture-historical school to report on my morning's activity. It was a good break. It isn't every day that I can work on a little bit of front-yard archaeology.

Our yard has always been a great site for playing archaeologist. The house is more than a hundred years old. Many people have lived there; the yard has seen many changes. Almost anywhere you dig you'll find some residuals of previous occupants, whether it be shards of glass or parts of toys. So, as I set out this morning to dig a hole for a banister next to the front stairs, I did not know whether I would find anything curious. This was another task that needed to be done to make the house eligible for decent home owner's insurance in due time.

In previous posts I have alluded to Howard Carter as a role model for me in archaeology. However, I need to clarify that his function for me is mostly confined to the context of encountering mummies. A few years ago I took a test on Facebook, "What kind of archaeologist are you?", and I came out as Flinders Petrie, a somewhat ambivalent honor. It's too bad that Dr. Henry Jones, Jr. is only a fictional character, and that he is never shown doing actual archaeology. I don't think he ever measured out a grid. And he really did not know that X frequently can mark the spot. It will do so all the time, if only you reverse the procedure: First you find the spot, then you paint an X on it. 

So, I had inserted the first post a few days ago. Since the hole was really close to the porch, I didn't expect to find anything, and I didn't. At that time, I had cut both poles, based on my calculations.

Skip this section if you hate numbers or equations.  The question was at what angle to cut the tops of the two posts to hold the rail properly. By measuring our neighbors' homemade banisters, it looked like 30 inches was about the proper height for a handrail. Since it had to be the same height along the stairs, that meant it had to be the same at ground level as well on the porch. So the back bottom post would have to be 30 inches plus 24 inches to sink into the ground, and the porch post would have to be the same height plus the height of the porch. The distance between the intended posts was also easy to measure. A is the distance between the posts; P1 and P2 are the two posts; H is the length of the handrail, which I couldn't measure directly. For the information I wanted, I realized that I only needed the dimensions of the top of the structure, which formed a right triangle. Now I could figure out the length of H, which also handily abbreviates for "hypotenuse," with the help of good old Pythagoras. Now I knew the dimensions of each line of the triangle, and I could call its three points X, Y, and Z (or anything else, such as Moe, Larry, and Curly, but I decided to be a little more conventional). Since it was a right triangle, it was obvious that angle ZXY = 90°. What I needed to know was the size of angle XYZ, so that I would know at what angle to cut the tops of the posts. I could do so by figuring out its cosine (the length of the side adjacent to the angle divided by the length of the hypotenuse):  A/H. I got my number and looked up on a table to what angle that number corresponded, and it was about 12°. Just to make sure, I did the sine on both angles, XZY and XYZ, as well as the cosine of XZY. The numbers all came out nicely at either 12° or 78°, so I was pretty confident that my trigonometry was holding up. Whether I could manage to saw the top of a 2x4 with that kind of exactness was another question.

 Now, to move back from the abstract to the concrete. People sometimes raise questions along the line of, "Just because you worked out the math on paper, why does that mean that it has to apply to reality?" The answer is (and we're not far from the issues in physics to which we will get back shortly), because that's how God made the world. God created numbers and their relationships as integral to the entire universe. The nineteenth century German mathematician Leopold Kronecker famously said that "God made integers, all else is the work of man." I really have to disagree with him. If pi is a constant found in nature, it's part of his creation. If complex numbers (x + iy) play a role in how the universe is put together, then God established those relationships. So, if I did the math correctly to an acceptable degree of accuracy or approximation, and my skill with a circular saw comes close enough to follow the numbers, the result must turn out in line with the calculations. I'm making a deal out of this because so many people nowadays have just given up on reason. We have a rational God, who created a rational universe, and there is nothing wrong with using our reason, as long as we use it correctly. (For example, know your multiplication table!)

So, I cleared my space and began putting the post hole digger to use. It's a very handy instrument; you can dig extremely deep holes with it without having to make the top get wider and wider. I was aiming for a depth of 24 inches.

At the nine-inch level, I made a startling discovery. There was a cord, right in the middle of my hole; it looked like a buried phone line--I didn't cut into it to identify its nature more closely. Such a line had no business being there; it had not been identified by the people who come and put little flags all over your yard when you call them before you dig. I doubt that I would have caused any harm if I had just snipped it. But whatever the nature of the cord, it probably didn't care about what I did or did not doubt.  So, to be sure I wasn't going to take any chances, I moved my excavation further out by a few inches and started over again. It turned out that I should have given it a couple of more inches because, when I reached the same level, there was that cord again, this time on the edge of the hole, but I could not proceed without hurting it. So, once again, I had to move further out and begin anew.

This was turning into quite a bit of additional exercise, and I hoped I could finish today. It was pretty hot, but then all of a sudden a nice breeze picked up, and the temperature fell by a few degrees, and I was able to move along nicely, which means with lots of breaks and being able to pick up again after the breaks.

My big concern was that due to the increase of the distance between the posts, my math calculations were obviously no longer going to apply as nicely. Would I be able to make adjustments that wouldn't make the whole thing look dorky?

At the depth of about one foot, I started to find a sizeable number of turquoise-colored pebbles, apparently providing evidence that the Turquoise-Colored Pebbles Culture had thrived here at some point. To this day one can find replicas of this very ancient civilization at the bottom of aquariums. I don't know whether modern science has an answer as to why I should have encountered these little artifacts in that precise location and at that depth. I collected a number of them, washed them with care, and you can see the result in the pictures. If any museum would like them, or if someone would like to use them for anthropological research, I'll be happy to make them available. (Please remember that I said "playing" archaeologist above.)

I continued digging to the required dimension, and then inserted the bottom pole. When I tried to balance the handrail on them, it turned out that my concern was well grounded. The angle was not going to work as tightly any longer. So, I adjusted the tops of both poles with a plane, and actually put some dirt back into the hole so that the bottom post was going to be right where it needed to be. Then I filled the larger part of the hole with cement. In our area, we get enough alternation between spells of dry days and heavy rain days in the summer that we can fasten posts with dry cement. The gradual curing  catalyzed by rain and ground water firms the cement better than the usual peanut-butter-consistent wet cement.

Having filled the hole, I needed to cut the 2 x 4 that would serve as handrail to a useful length. I inserted some 3-inch screws, and the basic banister was in place, adequate for inspection, I believe. I would like to add some things to it, but I have a few more similar tasks, so I may just let that one ride for the moment and take on the other projects first.

Win,

For the mathematically challenged among us, would not the following solution work as well.

1. Dig the post holes to the appropriate depth
2. Run a string from the top post to the bottom.
3. Mark the line along the posts
4. Remove the posts
5. Cut the posts
6. Replace the posts

Jimm, sure, as long as you manage to keep the posts exactly where they're going to be while you run the string, and keep the string taut. As I said to June when I sat down to do the math: "This is not really necessary; it just gives me a chance to implement the math I've been trying to keep alive all these years." But I also want to stress the idea that, quantum and relativity not-with-standing, God did create a "Pythagorean" universe, so to speak.

Win,

At least God created a world where formal relationships can objectively be described mathematically.

Once upon a time, I really enjoyed math, then I let it go to pot.

--jimm

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