I recently learned some things about how the young George Washington did math, including surveying. Mathematician and historian V. Frederick Rickey gave a talk 2 nights ago at the Mathematical Association of America here in DC, based on his study of GW’s “cypher books”, and I’d like to share a few things I learned.
(1) The young George appears to have used no trigonometry at all when finding areas of plots of land that he surveyed. Instead, he would ‘plat’ it very carefully, on paper, making an accurate scale drawing with the correct angles and lengths, and then would divide it up into triangles on the paper. To find the areas of those triangles, he would use some sort of a right-angle device, found and drew the altitude, and then multiplied half the base times the height (or altitude). No law of cosines or sines as we teach students today.
(2) He was given formulas for the volumes of spheroids and barrels, apparently without any derivation or justification that they were correct, to hold so many gallons of wine or of beer. (You probably wouldn’t guess that you had to leave extra room for the ‘head’ on the beer.) Rickey has not found the original source for those formulas, but using calculus and the identity pi = 22/7, he showed that they were absolutely correct.
(3) GW was a very early adopter of decimals in America.
(4 ) This last one puzzled me quite a bit. It’s supposed to be a protractor, but it only gives approximations to those angles. The results are within 1 degree, which I guess might be OK for some uses. I used the law of cosines to convince myself that they were almost all a little off. Here’s an accurate diagram, with angle measurements, that I made with Geometer’s Sketchpad.
His method was to lay out on paper a segment 60 units long (OB) and then to construct a sixth-of-a-circle with center B, passing through O and G (in green). Then he drew five more arcs, each with its center at O, going through the poitns marked as 10, 20, 30, 40, and 50 units from O. The claim is then that angle ABO would be 10 degrees. It’s not. It’s only 9.56 degrees.
I had a productive 24 hours!
- Night before last, I think I finally got Sky Wizard Digital Setting Circles installed on the 14″ alt-az telescope we were most generously donated by Alan Bromborsky. (That’s me, in the operations cabin at Hopewell Observatory, taking a break and a picture, long before completion.)
- So I went out to look at the sky at 1 AM. I saw no stars, but the 80% gibbous moon appeared to race dramatically through the clouds
- That afternoon, as I was driving out, I saw 5, maybe 6 tom turkeys playing hide-and-seek with me behind the trees. Believe me, they are REALLY GOOD at hiding behind little saplings, logs, and rocks! Or if you don’t believe me, ask anyone who’s tried to hunt them.
- Late that evening, I got a dry-ice-and-isopropanol particle detector working for the first time. (I had tried and failed, when I was a teenager, some 50 years ago, and failed several other times since then as well.) If you look at my little video, you can see the particles more easily than I could with your naked eye as I was filming it. Don’t ask me yet which ones are muons, which are alpha particles, and which are beta particles, because I don’t know yet. But you could look it up!
Productive 24 hours!
- I don’t trust any data on this, even Masters and Johnson, so any guesses on my part would be just that
- I’m only talking about having sex with another person, either Penis-In-Vagina or any other sort of sexual activity, which I am not about to list here — use your own imagination if you want to.
- And no, I’m not going to tell you anything about my own habits or those of anybody else I know. However, if you want to tally up your OWN time doing ‘that’, feel free.
A long time ago I had a course by Prof Jim Sandeful of Georgetown U and Dr Monica Neagoy on teaching with “discrete math” — very useful and interesting stuff that often does not get discussed in the standard American curriculum. I enjoyed it a lot.
Among other topics, I decided to write a little computer program that would model exponential random decay of radioactive elements. (Iirc I did this in Pascal and in BASIC, on the C-64, IBM-PC, Apple II, and Commodore Amiga. That was fun.)
One subtopic that came up, but which I never figured out how to model, was how to describe the frequency of some trait (eg red hair, striped tail, or growing a third eye…) in a population. I had long thought about how to do that but not until today did I begin to make some progress, so please allow me to share.
I’m going to make up a very-much simplified example using a Punnett square, something like this:
Upper-case B and lower-case b in this diagram stand for two different versions of a particular (but mostly imaginary) gene that controls whether a person has blue eyes or brown. In this hypothetica example, the upper-case B gene causes brown eyes and is dominant, where the lower-case “b” causes eyes to be blue and is recessive. Thanks to the magic of sexual reproduction, you get two copies of each gene, 1 from Mom and one from Dad, whether they stick around and raise you or not. (You have two similar-but-not-identical copies of each chromosome except for the X and Y chromosomes; your two versions of each gene are carries in corresponding locations on each of the two chromosomes. If I got this right.)
If you have brown eyes, then your genes might be BB or they might be Bb or bB (same thing). If you have blue eyes, then you have bb genes for sure — again, in this hypothetical scenario.
This Punnett square shows the probability of what will happe if two parents who carry Bb genes have sex and produce offspring. It reminds me very much of how we use an area model to show that (X + Y)*(X + Y) equals X^2 + 2*X*Y + Y^2.
In any case, each of those parents carries Bb genes, and when the eggs and the sperm cells are manufactured inside the parent’s ovaries and testes, one or the other version of the gene is put inside, but not both. And it’s random. So since each parent has a Bb gene, its probability of passing along upper case B (brown) is 1/2 or 50%, as is the probability of passing along lower case b (blue eyes).
You can now find the probability of all of the outcomes shown in the interior of the diagram. The upper left hand corner is BB, pure brown eyes, with probability 1/4 because 1/2*1/2=1/4 and also in this case all of the sections really do have equal areas.
The upper right hand and lower left hand corners represent the Bb cross; the child will have brown eyes. The probability of a Bb cross is 1/4 plus 1/4, or 1/2.
The lower right hand corner is the region representing the probability of pure bb offspring which have (recessive) blue eyes. The probability of bb is 1/4.
Now let us add a couple of features.
1. This is not just a single mom-dad pairing: this is a representation of an entire reproducing population where genes B and b are present, each 50% of the time.
2. Let us also pretend that the bb combination is fatal: not a single one of them survive to adulthood and to leave offspring. (This is a very extreme hypothetical example of how evolution operates. Normally Deleterious genes aren’t so uniformly fatal!) or alternatively, a breeder of plants or animals might decide to not permit any of the blue-eyed bb offspring to reproduce. Eugenicists used to advocate sterilizing anyone who exhibited harmful, recessive genes,in order to improve the remainder of the human race.
At first glance, You would think that this sort of genetic selection, either by artificial or natural means, would work very quickly, and that after just a few generations, the proportion of the population that was blue-eyed would vanish.
Jim Sandefur said no, it would take a really long time. I forgot the details, and just worked them out today. I’ll work out the details for you later when I have a larger screen. But:
Bottom line: even with this 100% culling of recessive genes, the proportion of blue eyes goes down as the harmonic series (1/X), where X is 4, then 5, 6, 7, 8, etc
So if the first generation has 1/4 (25%) blue eyes, and if every single individual with blue eyes is somehow prevented from reproducing, then the next generation will still carry the lower-case b gene 1/5 (20%) of the time.
And if children with blue eyes (bb) are still prevented from reproducing, the third generation will still pass on the lower-case b gene one-sixth, or 16.67% of the time, and the next generation will pass on the lower-case b gene one-seventh (14.29%) of the time. The next generation passes on b genes one-eighth (12.50%) of the time, then one-ninth of the time (11.11%), then one-tenth of the time (10.00%) and so on. At first the decrease is pretty rapid, but after that it slows to a craw, and the world would never be entirely free of the pure bb. After 100 generations, there still would be 1/103 (almost 1%) of the population carrying genes that can pass on blue eyes.
At 25-30 years for a human population to reproduce, you are talking about 2,500 to 3,000 years!
However, the fraction of the population that actually is born with blue eyes apparent to everybody will fall much faster. The proportions would be 1/4 in the initial generation, followed by 1/9, then 1/16, then 1/25, then 1/36, then 1/49, then 1/64, and so on, with the ratio being 1/X^2 (one over x-squared) rather than 1/x.
So, by 10 generations, under this hypothetical, 100%-effective sterilization or extermination regime, the proportion of the population with visible blue eyes would have fallen to 1/169, about six-tenths of a percent. However, the fraction of the population that still carries the genes for blue eyes would remain at 1/13 of the population, about 7.7% of the total.
However, perhaps conditions might flip-flop. In my hypothetical problem here, perhaps the conditions making blue eyes fatal would disappear after a number of generations. (Even if Hitler’s nasty 1000-year Reich would not have been enough to eradicate whatever enemy genes!) In fact, perhaps the reverse would be true: having brown eyes would be a fatal handicap under some conditions. Then the prevalence of blue eyes would rise to the fore in their place, but there would be an enormous die-off of all those who had brown eyes, which would mean the vast majority of the population. So all that would be left would be those formerly recessive genes, and the formerly dominant genes would be wiped out completely.
More realistically: recessive genes that make people susceptible to die from some particular disease or parasite or environmental factor do definitely get reduced in frequency over time, as I hope I have shown. However, they do not disappear completely for a very long, long time (if ever!) unless the entire population is reduced to just a handful of individuals, none of whom carry that gene, just by chance.
Evolution does work on those time scales. Human societies and any proposed eugenics program do not. Evolution has no direction, and is essentially blind, like a mathematical algorithm.
People often say that everything happens for a reason. Often, that reason is simply the laws of probability, which are extremely hard for most people to handle. Myself included.
I’ve been helping students at the First Light Saturday science school at the Carnegie Institution for Science for several years now. We’ve done a variety of activities, from making small generators and exploring water power; building and programming robots; measuring the chemical content of foods; growing plants under various conditions (including simulated zero-gravity); and this year, experimenting with light, including building their own small telescopes.
They so far have made three such telescopes: a Galilean, a Keplerian, and a more modern achromatic refractor. Here is what they used to make them. The lenses, all from Surplus Shed, cost a grand total of Five dollars per set. The PVC was a bit less, I think.
Because of bad weather, our winter term was somewhat shortened. Here is one example of what they will finish – a small refractor on a tripod! (They’ll need to supply their own cat, though…)
VANTA black is apparently the blackest material in the world – so far. Made out of vertically-oriented arrays of carbon nanotubes grown in situ, reflects only something like 0.035% of the light that hits it. Imagine that coating the inside of a telescope tube! No more stray reflections!
If you have ever cleared a sidewalk after a snowstorm (like I did this morning), you’ve probably noticed that shoveling snow is a lot of hard work.
I wondered just how hard I was working to shovel our porch and sidewalk, so I did some rough calculations.
Not knowing the weights or masses of snow or water in American customary units I did it all in metric units because it’s so much easier.
Using a construction tool, I measured the snow as being about 13″ deep, or about 33 cm (1/3 of a meter). I shoveled a path that was roughly a meter or so wide, and a grand total of about 21 long paces (roughly a meter each) in length.
Which means I had shoveled a volume of 1/3 *21*1 or 7 cubic meters. If that was all liquid or solid water, that would be exactly 7 metric tons. But snow is about 90%air, so if we divide that by 10, we get 700 kilograms instead, or about 1500 pounds of fluff.
Huff, huff, puff indeed.
By the way, my son Josef Brandenburg, a DC-area fitness expert and personal trainer, has a nice interview with Bruce Depuyt on the right way to shovel so that you don’t throw your back out and end up in the emergency room along with many thousands of other folks. (I didn’t.)