Humans had ZPG for 5,000 years!!!

I’m listening to “Against the Grain: A Deep History of the Earliest States”, a new book exploring recent findings on the 97% of our past as human beings before writing, tax collectors, cities, and modern states.

One surprising finding: from ten thousand BC to five thousand BC, the total population of us human beings (homo sapiens sapiens) world wide went from about 4 million souls to 5 million — according to one estimate by Macevedy & Jones. (There are other estimates: )

That was rather slow growth, the author noted. But *how* slow, this retired math teacher wondered?

So I got out a pencil and my notebook and wrote an equation. I used G for the annual growth factor, and wanted to see how close to 1.0000000 it was.

(Note: If G is exactly 1, then the population never changes; if G is less than 1, the population shrinks with exponential decay. If G=2.0, then the population doubles every year, which obviously can’t happen in any human population anywhere nor at any time. Though it certainly can for some of our commensal pests like mice…)

So, Macevedy & Jones’ initial population estimate of 4,000,000 (assuming smooth exponential growth over five millennia — a useful mathematical fiction) gets multiplied by G, whatever that might be, five thousand times (ie by G raised to the 5,000th power) to produce 5,000,000 people.

Or, 4000000*G^5000 = 5000000

Dividing both sides by four million I get

G^5000 = 1.25

The only way I know to solve that is to take the logarithm of both sides. Doing that with base ten and using the special laws of logs, I get

5000*log (G) = log (1.25)

Then I divide both sides by 5000 and I get

Log(G) = log(1.25) /5000

Then I exponentiate both sides using the original log base (ten), and I get

G =10^( log(1.25) / 5000)

At this point I use a calculator on my phone, typing in exactly the stuff on the RH side of the equals sign. And I get

10^(log10(1.25)/5000) = 1.00004463

Which is very, very close to unity. How close? Let us subtract one from that. We get

0.0000463 or 4.463e-05 in scientific notation. Or roughly 45 parts in a million. Mind you, there were a grand total of four million of our ancestors on the planet then, so we can multiply that 45 by four, and we get 180.

But what does that mean?

It means that on average, out of the ENTIRE HUMAN POPULATION ON THE PLANET AT THAT TIME, there was a net increase of people of only 180 souls per year.

That’s all.

On the whole planet!!!!

They had nearly achieved zero population growth!

But during the next five thousand years our population really exploded, to some hundreds of millions of people. Doinfg the same calculation, I found that the annual growth rate was about 1.00074, or 0.074%, or 74 additional net humans per year per hundred thousand, or about 74 thousand net new humans per year total, world-wide, once they got up to about a hundred million people.

That’s just up to the year 0 BC/AD.

Let us remember always that this planet right here is the only one we humans can possibly live on or get to in any numbers. We are as a species have done incalculable damage. Here in North America, think of the thoughtless and greedy extermination (or near-extermination) of the passenger pigeon; the American chestnut, elm, hemlock and ash; the buffalo; almost all of old-growth forests; most anadromous Atlantic fish; and Chesapeake bay oysters — all of which used to be plentiful beyond belief.

Some species are now recovering, such as deer, beavers, skunks, rabbits, foxes and coyotes.. Why is that? If you look at photos of Virginia countryside from 90 to 150 years ago, you see very, very few trees. Lumber companies and plantation owners and small farmers had cut them all down to plant grain and cash crops. Plowed land erodes quickly from both wind and rain. Those formerly fertile fields became uneconomical to farm, and so field after field (including ones I played or worked or hunted on as a kid and young man) have been allowed to regrow brush and then trees or housing developments, shopping centers, and pavement. So East of the Mississippi, there has been a dramatic increase in percentage of tree canopy over the last century.

However, some countries are repeating America’s mistakes and are cutting down primeval firsts as fast as they can…

Published in: on October 6, 2022 at 10:20 am  Comments (2)  
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Perhaps a slight downward trend in new COVID cases?

Prompted by a former colleague, I did some tedious work at the CDC site on the numbers of COVID-19 cases each day, going back to January. I found what looks like a weekly up-and-down oscillation pattern that might have to do with whether offices are open and whether reports are made promptly, or might have to be delayed until the end of the weekend. However, it does appear to me that there might be a slow, but real, downward trend over the last few weeks — mostly because the vast majority of us are practicing self-isolation. Here is the graph I made:

new covid cases in the US, per day

Clearly, we are no longer seeing either a steady increase in the number of new cases each day as we were seeing from week 6 to week 10 nor (God forbid!) exponential growth as we were seeing back in March. If we were having exponential growth, it would show up as a horizontal line in the graph below.

daily rate of increases

However, if we stop the social distancing, if we all stop wearing masks and washing hands, if we all start going to movies and restaurants and museums and bars as if this is all over, and if kids go play on playgrounds and go back to school as normal, then exponential growth will raise its ugly, feverish head, and perhaps millions will die.

By the way, I cannot easily find equivalent data on the CDC website for daily deaths; just new diagnosed cases. The COVID death data may be there, but it’s really difficult to dig out. Maybe someone has a source?

The Pandemic Is Far From Over

While the rate of increase per day in the number of deaths is generally down, the COVID-19 pandemic is far from over. In general, more people are still dying each day in the US from this disease than the day before, as you can see from this data, which is taken from the CDC. The very tall bar on day 27 is when New York City finally added thousands of poor souls who had in fact died from this virus. (Day 27 means April 9, and Day 41 means April 30, which is today.)

Opening up the economy and encouraging everybody to go back to work, play, and school will mean a rebirth of exponential growth in deaths and in diagnosed cases after about 2 weeks, since this disease takes about that long to be noticed in those who have been exposed. And once everybody is back on the streets and in the stores and schools, the disease WILL spread exponentially. Opening wide right now, when we still can’t test or follow those who may be infected, would be a huge mistake.

us covid deaths per day

Only somebody as clueless as our current Grifter-In-Chief and his brainless acolytes could be recommending something so irresponsible, against the advice of every medical expert. Maybe they think that only the poor, the black, and the brown will get this disease. Wrong.

The shutdown, while painful, appears to have saved a LOT of lives so far

If you recall, the growth of the new corona virus disease in the US (and many other countries) at first looked to be exponential, meaning that the number of cases (and deaths) were rising at an alarming, fixed percent each and every single day.

Even if you slept through your high school or middle school math lessons on exponential growth, the story of the Shah and the chessboard filled with rice may have told you that the equation 2^x gets very, very hairy after a while. Pyramid schemes eventually run out of suckers people. Or perhaps you have seen a relatively modest credit-card bill get way out of hand as the bank applies 8 percent interest PER MONTH, which ends up multiplying your debt by a factor of 6 after just 2 years!

(If the total number of deaths were still increasing by 25 percent per day, as they were during the middle of March, and if that trend somehow continued without slowing down, then every single person residing inside America’s borders would be dead before the end of May. Not kidding! But it’s also not happening.)

However, judging by numbers released by the CDC and reported by my former colleague Ron Jenkins, I am quite confident that THE NUMBER OF CASES AND DEATHS FROM COVID-19 ARE NO LONGER following a fixed exponential curve. Or at least, the daily rate of increase has been going down. Which is good. But it’s still not zero.

Let me show you the data and fitted curves in a number of graphs, which often make complex things easier to visualize and understand.

My first graph is the total reported number of deaths so far in the US, compared to a best-fit exponential graph:

Deaths in US are not growing exponentially

During the first part of this pandemic, during the first 40 or so days, the data actually fit an exponential graph pretty well – that is, the red dotted line (the exponential curve of best fit) fit the actual cumulative number of deaths (in blue). And that’s not good. However, since about day 50 (last week) the data is WAY UNDER the red dots. To give you an idea of how much of a victory that is: find day 70, which is May 9, and follow the vertical line up until it meets the red dotted line. I’ll wait.

Did you find it? If this pandemic were still following exponential growth, now and into the future, at the same rate, we would have roughly a MILLION PEOPLE DEAD BY JUNE 9 in just the US, just from this disease, and 2 million the week after that, and 4 million the next week, then 8 million, then 16 million, and so on.


As you can see — the blue and red graphs have diverged. Ignore the relatively high correlation value of 0.935 – it just ain’t so.

But what IS the curve of best fit? I don’t know, so I’ll let you look for yourself.

Is it linear?

Deaths in US are not growing in a linear fashion

This particular line of best doesn’t fit the data very well; however, if we start at day 36 or thereabouts, we could get a line that fits the data from there on pretty well, like so:

maybe this purple line


The purple line fits the blue dots quite well after about day 37 (about April 6), and the statistics algorithms quite agree. However, it still calls for over 80,000 Americans dead by May 8. I do not want the slope of that line to be positive! I want it to turn to the right and remain horizontal – meaning NOBODY ELSE DIES ANY MORE FROM THIS DISEASE.

Perhaps it’s not linear? Perhaps it’s one of those other types of equations you might remember from some algebra class, like a parabola, a cubic, or a quartic? Let’s take a look:

Deaths might be growing at a 2nd degree polynomial rate - still not good

This is a parabolic function, or a quadratic. The red dots do fit the data pretty well. Unfortunately, we want the blue dots NOT to fit that graph, because that would, once again, mean about a hundred thousand people dead by May 8. That’s better than a million, but I want the deaths to stop increasing at all. Like this piecewise function (which some of you studied). Note that the purple line cannot go back downwards, because generally speaking, dead people cannot be brought back to life.

maybe this purple line - nah, prefer horizontal

Well, does the data fit a cubic?

deaths fit a cubic very well

Unfortunately, this also fits pretty well. If it continues, we would still have about a hundred thousand dead by May 8, and the number would increase without limit (which, fortunately, is impossible).

How about a quartic (fourth-degree polynomial)? Let’s see:

4th degree polynomial is impossible - people do NOT come back to life

I admit that the actual data, in blue, fit the red calculated quartic red curve quite well, in fact, the best so far, and the number of deaths by Day 70 is the lowest so far. But it’s impossible: for the curve to go downwards like that would mean that you had ten thousand people who died, and who later came back to life. Nah, not happening.

What about logarithmic growth? That would actually be sweet – it’s a situation where a number rises quickly at first, but over time rises more and more slowly. Like this, in red:

logarithmic growth

I wish this described the real situation, but clearly, it does not.

One last option – a ‘power law’ where there is some fixed power of the date (in this case, the computer calculated it to be the date raised to the 5.377 power) which explains all of the deaths, like so:

no sign of a power law

I don’t think this fits the data very well, either. Fortunately. It’s too low from about day 38 to day 29, and is much too high from day 50 onwards. Otherwise we would be looking at about 230,000 dead by day 70 (May 8).

But saying that the entire number of deaths in the US is no longer following a single exponential curve doesn’t quite do the subject justice. Exponential growth (or decay) simply means that in any given time period, the quantity you are measuring is increasing (or decreasing) by a fixed percentage (or fraction). That’s all. And, as you can see, for the past week, the daily percentage of increase in the total number of deaths has been in the range of three to seven percent. However, during the first part of March, the rate of increase in deaths was enormous: 20 to 40 percent PER DAY. And the daily percent of increase in the number of cases was at times over A HUNDRED PERCENT!!! – which is off the chart below.

daily percentages of increases in covid 19 cases and deaths, USA, thru April 25

The situation is still not good! If we are stuck at a daily increase in the number of deaths as low as a 3%/day increase, then we are all dead within a year. Obviously, and fortunately, that’s probably not going to happen, but it’s a bit difficult to believe that the math works out that way.

But it does. Let me show you, using logs.

For simple round numbers, let’s say we have 50,000 poor souls who have died so far from this coronavirus in the USA right now, and that number of deaths is increasing at a rate of 3 percent per day. Let’s also say that the US has a population of about 330 million. The question is, when will we all be dead if that exponential growth keeps going on somehow? (Fortunately, it won’t.*) Here is the first equation, and then the steps I went through. Keep in mind that a growth of 3% per day means that you can multiply any day’s value by 1.03, or 103%, to get the next day’s value. Here goes:

in 10 months we are all dead

Sound unbelievable? To check that, let us take almost any calculator and try raising the expression 1.03 to the 300th power. I think you’ll get about 7098. Now take that and multiply it by the approximate number of people dead so far in the US, namely 50,000. You’ll get about 355,000,000 – well more than the total number of Americans.

So we still need to get that rate of increase in fatalities down, to basically zero. We are not there yet. With our current highly-incompetent national leadership, we might not.


* what happens in cases like this is you get sort of an s-shaped curve, called the Logistic or logit curve, in which the total number levels off after a while. That’s shown below. Still not pleasant.

I have no idea how to model this sort of problem with a logistic curve; for one thing, one would need to know what the total ‘carrying capacity’ – or total number of dead — would be if current trends continue and we are unsuccessful at stopping this virus. The epidemiologists and statisticians who make models for this sort of thing know a lot more math, stats, biology, and so on than I do, but even they are working with a whole lot of unknowns, including the rate of infectiousness, what fraction of the people feel really sick, what fraction die, whether you get immunity if you are exposed, what is the effect of different viral loads, and much more. This virus has only been out for a few months…

logistic curve again


What’s the best approach – should we lock down harder, or let people start to go back to work? Some countries have had lockdowns, others have not. How will the future play out? I don’t know. I do know that before we can decide, we need to have fast, plentiful, and accurate tests, so we can quarantine just the people who are infected or are carriers, and let everybody else get back on with their lives. We are doing this lockdown simply because we have no other choice.

COVID-19 Numbers in the US do not seem to be growing exponentially

Looking at the past month of CDC-reported infections and deaths from the new corona virus, I conclude that there has been some good news: the total number of infections and deaths are no longer following an exponential growth curve.

The numbers are indeed growing, by either a quadratic (that is, x^2) or a quartic (x^4) curve, which is not good, and there is no sign of numbers decreasing.

BUT it looks as though the physical-social distancing and self-quarantining that I see going on around me is actually having an effect.


Here is my evidence: the actual numbers of infected people are in blue, and the best-fit exponential-growth equation is in red. You can see that they do not match well at all. 

total cases US not looking exponential

If they did match, and if this were in fact exponential growth, we would have just about the entire US population infected by the end of just this month of April – over 300 million! That no longer seems likely. Take a look at the next graph instead, which uses the same data, but polynomial growth:

total cases US looking second power

Just by eyeballing this, you can see that the red dots and blue dots match really, really well. When I extend the graph until the end of April, I get a predicted number of ‘only’ 1.5 million infected. Not good, but a whole lot better than the entire US population!

Also, let’s look at total cumulative reported deaths so far. Here are the CDC-reported numbers plotted against a best-fit exponential curve:

deaths do not seem to be exponential

Up until just a few days ago, this graph was conforming pretty well to exponential growth. However, since about April 8, that seems to be no longer the case. If the total numbers of deaths were in fact growing at the same percentage rate each day, which is the definition of exponential growth, then by the end of April we would have 1.5 million DEAD. That’s THIS MONTH. Continued exponential growth would have 1.2 BILLION dead in this country alone by the end of May.

Fortunately, that is of course impossible.

Unfortunately all that means is that the virus would run out of people to infect and kill, and we would get logistic growth (which is the very last graph, at the bottom).

death seem to be 4th power polynomial

This fourth-degree mathematical model seems to me to work much better at describing the numbers of deaths so far, and has a fairly good chance of predicting what may be coming up in the near future. It’s still not a good situation, but it shows to me that the social and physical distancing we are doing is having a positive effect.

But let’s not get complacent: if this model correctly predicts the next month or two, then by the end of April, we would have about 60 thousand dead, and by the end of May we would have 180 thousand dead.

But both of those grim numbers are much, much lower than we would have if we were not doing this self-isolation, and if the numbers continued to grow exponentially.


FYI, a logistic curve is shown below. Bacteria or fungi growing in a broth will grow exponentially at first, but after a while, they not only run out of fresh broth to eat, but they also start fouling their own environment with their own wastes. WE DO NOT WANT THIS SITUATION TO HAPPEN WITH US, NAMELY, THAT WE ALL GET INFECTED!!!

logistic curve again

Various graphs for deaths from COVID-19, so far

I wrote that I would show you what various graphs of various types of simple models look like for deaths so far due to the current corona virus: linear, exponential, polynomial, and so on. I think that a fourth-degree (not third-degree, like I wrote earlier) seems to fit the data best so far, and that’s better than exponential growth.

First, let’s look at a straight-line best-fit model, superimposed by Excel on the data. (Note: deaths are on the Y, or vertical, axis; the X-axis represents days since the beginning of March. So today, the 6th of April, is day 37 (31 + 6). The dotted red line represents the line of best fit, and the blue dots are the CDC-announced numbers of deaths so far.

is it linear

As you can see, the straight dotted line doesn’t fit the data very well at all. R-squared, known as the correlation coefficient, tells us numerically how well it fits. If R or R^2 equals 1.000, then you have absolutely perfect correlation of the data to your model. Which we do NOT have here. By the way, in that model, then by mid-June we would have about 22,000 dead from this disease.

OK, let’s look at an exponential curve-of-best fit next:

is it exponential

As you can see, this red curve fits the data a LOT better, and R-squared is a lot higher.


We do NOT WANT EXPONENTIAL GROWTH OF THIS OR ANY OTHER DISEASE, BECAUSE IT MEANS WE ALL GET IT! In fact, if this model is accurate and isn’t slowed down, then by mid-June, just plugging in the numbers, we would have 3.3 BILLION (not million) people dead in the US alone. Fortunately, that won’t happen.

BUT there are some parts of the data where the curve doesn’t fit perfectly — let me point them out:

is it exponential -2

At the upper right-hand end, the red dotted line is quite a bit higher than the blue dots. Fortunately. And near the middle of the graph, the blue dots of death are higher than the red line.

OK, let’s look at some polynomial models instead:

is it a second degree polynomial

This is a fancy version of the simple y=x^2 parabolas you may have graphed in Algebra 1. Once again, this doesn’t do a terrific job of conforming to the actual data. At the right-hand end, the blue dots of death are higher than the curve. In addition, if we continued the red curve to the left, we would find that something like two thousand people had already died in the US, and presumably came back to life. Which is ridiculous.

However, if this model were to hold true until mid-June, we would have 127 thousand dead. Not good.

Let’s try a third-degree polynomial (a cubic):

is it a third degree polynomial

That’s pretty remarkable agreement between the data and the equation! That’s the equation I was using in my earlier post. The R-squared correlation is amazing. Unfortunately, if this continues to hold, then we would have about 468 thousand dead in the US.

Let’s continue by looking at a fourth-degree polynomial curve fitted to the data:

is it a fourth degree polynomial

That is an amazingly good fit to the data! Unfortunately, let’s hope that it won’t continue to fit the data, because if it does, then we are looking at a little over a MILLION dead.

Let’s hope we can get these totals to level off by physically distancing ourselves from other households, washing our hands, and getting proper protective garments and testing technology to our medical personnel.


Here’s another model that unfortunately does NOT work: logarithmic growth. If it were the case, then we would have about 10,700 deaths by mid-June.

is it logarithmic


“And forgive our debts, as we forgive those who owe us!”

The title of this post might remind you of part of the so-called Lord’s Prayer, which in English is usually rendered “And forgive us our trespasses, as we forgive those who have trespassed against us.”

This sounds like forgiving sins, but in Latin, which I studied for about six years, the prayer is really about forgiving debts:

“et dimitte nobis debita nostra sicut et nos dimittimus debitoribus nostris”

I don’t know enough Greek to be able to comment on the original meaning of the words as apparently written down in the New Testament in that language, but it is generally accepted that Jesus (if he really existed) spoke Aramaic – but only a few of his (alleged) words were recorded in that language, since the entire NT was written in Greek, not in Hebrew or Latin, and definitely not in English!

The following book makes the argument that forgiving debts, wholesale, was essential if you wanted to avoid stratification of society into a class of oligarchs and a class of everybody else, who were essentially little better than slaves. They make the point that compounded interest grows exponentially and without limit, but economic growth does NOT: it follows a logistic curve at best, which means that there are certain limits.

For example, while bacteria growing in a petri dish appear to grow exponentially for some hours, perhaps for a few days, eventually, there is no more uncontaminated agar for the bacteria to eat, and they start drowning in their own waste products. So despite what one learns in most Algebra classes (including my own), bacterial growth is in actually logistic, not exponential. However, unless debt is periodically forgiven – which seldom if ever happens these days – the debtors end up drowning in debt, as you might be able to discern from this little graph I made:

logistic versus exponential growth

I haven’t read the book, but the review is most interesting. Here is a quote:

Nowhere, Hudson shows, is it more evident that we are blinded by a deracinated, by a decontextualizedunderstanding of our history than in our ignorance of the career of Jesus. Hence the title of the book: And Forgive Them Their Debts and the cover illustration of Jesus flogging the moneylenders — the creditors who do not forgive debts — in the Temple. For centuries English-speakers have recited the Lord’s Prayer with the assumption that they were merely asking for the forgiveness of their trespasses, their theological sins: “… and forgive us our trespasses, as we forgive those who trespass against us….” is the translation presented in the Revised Standard Version of the Bible. What is lost in translation is the fact that Jesus came “to preach the gospel to the poor … to preach the acceptable Year of the Lord”: He came, that is, to proclaim a Jubilee Year, a restoration of deror for debtors: He came to institute a Clean Slate Amnesty (which is what Hebrew דְּרוֹר connotes in this context).

So consider the passage from the Lord’s Prayer literally: … καὶ ἄφες ἡμῖν τὰ ὀφειλήματα ἡμῶν: “… and send away (ἄφες) for us our debts (ὀφειλήματα).” The Latin translation is not only grammatically identical to the Greek, but also shows the Greek word ὀφειλήματα revealingly translated as debita: … et dimitte nobis debita nostra: “… and discharge (dimitte) for us our debts (debita).” There was consequently, on the part of the creditor class, a most pressing and practical reason to have Jesus put to death: He was demanding that they restore the property they had rapaciously taken from their debtors. And after His death there was likewise a most pressing and practical reason to have His Jubilee proclamation of a Clean Slate Amnesty made toothless, that is to say, made merely theological: So the rich could continue to oppress the poor, forever and ever. Amen.

Share this video with 5 friends. 

You need to view and share this John Oliver video.


Yes, share it with 5 people and watch the power of exponential growth — uh, pyramid schemes — like HerbaLife or Amway.

This entertaining but informative video shows how modern-day pyramid schemes work, but it doesn’t explain how they keep getting away with it.

If you join one of these MLMs, or pyramids, you make nothing by selling the product. You only earn by signing up new sales-people, who give you money. But to get the right to do that you have to buy a lot of product, which often ends up in garages or closets.

And here’s the kicker: if you actually do everything right as per the recommendations of the MLM scheme, you run out of human beings. This is classic exponential growth: the founder gets 5 salespeople to sell under him. Each one of them gets 5 more so that generation is 25 people. Then each of those gets 5, for 125, each time multiplying by 5.

I just had my calculator compute 5^10, which means 10 generations, which really doesn’t sound like that many levels,  and I got nearly 10 million people, larger than quite a few states in the USA. Ten generations of friends and coworkers and family getting friends and  coworkers and family members to sign up should only take a few months, right, at about a week per generation of suckers salespeople?

Then: After 15 generations my calculator says, uh, I’m having a hard time counting all the decimal places, because it reads 3.051758713E10, also known as 30,517,578,125.

Of course, that’s impossible. Most folks who get sucked into the MUlti-Level Marketing morass actually earn nothing at all — even by the companies’ own admissions. (See the video!) but they all buy a LOT of product, benefiting the people above them.but I think that means a bit over 30 BILLION people, with a B. Now, last time I checked, we only have about 7.4 billion human beings alive today. So, each and every single Homo sapiens on all seven continents or even sailing on the ocean or living on a remote island somewhere, would have to be a distributor/salesperson/supervisor/sucker, about four times over, just on level 15. If we add in all of the  people from levels 1 through 14, we actually get about 38 BILLION suckers customers dealers.

Why are these things still allowed? Is it because they have paid off some important legislators? On the face of it, pyramid schemes (think Ponzi) are all illegal. How do these companies continue to operate and prey on folks who dont know better?

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