The stars at night are big and bright. As advertised in last month's diary, the Mrs and I spent the last week of March and first weekend of April on vacation in Texas. Some friends have bought a ranch in the Hill Country there, a few miles outside Fredericksburg. We spent a convivial couple of days with them at the ranch then went off exploring in our rental car not from Avis.
By happy coincidence, the March 18th issue of The Economist, which arrived the day before we left home, had a special report on Texas. It was pretty darn positive.
In the 12 months starting in July 2021 Texas added a net 471,000 residents, the most of any state. That is equivalent to 1,290 new Texans each day, or around 9,000 a week. Only a quarter of those came from natural increase (the difference between births and deaths). Another quarter were immigrants from abroad. But roughly half were arrivals from elsewhere in America …
Texas has many attractions. The state government does not levy any income tax. The cost of living is 7.5 percent below the national average. A big part of that is housing, which is 15 percent cheaper than the national average. That makes the typical home in Texas 56 percent cheaper than the equivalent in California. The state generates jobs even faster than it attracts people: employment grew by 28 percent between 2010 and 2021.
Hoo-ee … or rather Yee-haw! And then this:
The legislature meets for 140 days every other year, far less than in most states, yet wags joke that Texans would prefer them to meet for two days every 140 years.
I'm with the wags on that. We already have far too many laws. Most of what our legislators do is unnecessary when not actually wicked; most of it is done to appease wealthy donors and aggressive public-sector employee lobbies masquerading as "unions". And the ruling class doesn't respect our laws anyway. How much respect has the current administration shown for our immigration laws?
So we boarded the plane to Texas in a positive mood. Texas did not disappoint us.
The Lone Rock State. Sunday, March 26th. Driving out across the Hill Country to our friends' ranch Saturday night we noticed an oddity of the landscape. Every so often, at a dip in the road, there would be an upright orange wooden post with horizontal marks indicating height from the ground: 1 ft., 2 ft,. 3 ft., … all the way up to six or seven feet. Wha?
Our friends, when we arrived at the ranch, explained. The Hill Country is one humongous slab of solid rock with only a thin covering of soil. When there's heavy rain the water, with no deep soil to soak it up, just rolls down the rock to the nearest low point … and stays there. The orange post tells you how deep the water is where it's settled.
We got some more evidences of this theory when we went exploring. Meanwhile there was ranch life to enjoy and burros to be fed.
The burros came with the ranch. The matter of their health came up in one conversation. I said I supposed that if one died he'd be buried on the ranch. Maybe not, said our host. Hard to bury anything that size. You dig down a few inches, you hit rock.
The Lone Rock state! However do they build anything?
A superb museum. Monday, March 27th. We explored Fredericksburg.
The town still has a German vibe, 177 years after being founded and named after a Prince of Prussia, a great-grandnephew of Frederick the Great. As a long-time Germanophile, that put me in a good mood right away. We had an excellent meal at a German restaurant: Wurst, Butterkartoffeln, Sauerkraut, the whole deal.
Fredericksburg gets German tourists, too. While I was browsing the books for sale in the lobby of the Vereins Kirche Museum a middle-aged lady came in, went to the desk, and asked the curatrix, in accented English, if she could speak German. She could.
After some exchanges the curatrix stepped away to fetch something. I took the opportunity to socialize with the visitor using my schoolboy German.
Me: "Where in Germany are you from?"
She (obviously pleased to meet yet another German-speaker): "From Hanover."
Me: "Ah. I was born in England. We had a king from Hanover."
She (smiling mischievously): "More than one, I think."
All right, it wasn't conversation at the highest intellectual level; but it's been a lot of years, dammit. And these good-natured microscopic exchanges all help oil the machinery of international relations.
Chester Nimitz, the great naval Commander in Chief of WW2, was born and raised in Fredericksburg of German ancestry. The town now boasts the National Museum of the Pacific War, located in what was once a hotel belonging to the admiral's grandfather and known locally as the Nimitz Museum. We spent a very happy and instructive afternoon browsing around in there.
What was most instructive to me was being reminded of the sheer number of battles in the Pacific theater. Guadalcanal, Tarawa, Kwajalein, Eniwetok, Saipan, Guam, Tinian, … Good grief! And there was of course much grief for the loved ones of thousands of Americans who died fighting in those battles. May we never forget them.
Tremendous historical events, very well illustrated and described, some of them with sound effects. This is a superb museum. For Mrs Derbyshire's interest there are also very good presentations of events preceding the war, way back into Japanese and Chinese issues in the 19th century. (Although one of the explanatory boards mis-spells the name of the guy who led the Taiping rebellion: it was Hong Xiuquan, not Hong Xinquan.)
Wokesters stonewalled. Tuesday, March 28th. Off into the Hill Country exploring. First stop: LBJ's birthplace and ranch.
My opinion of Texas and its citizens went up another couple of ticks on learning that the nearest town to the LBJ homestead is still named Stonewall in tribute to the Confederate general. Haven't the Red Guards tried to get it renamed? If they have, they haven't succeeded. Score one for sanity.
The homestead, presidential museum, and ranch are all pleasant enough, with lovely landscapes all around.
LBJ was the Antichrist in my mid-1960s lefty-student days, though. Now, in my early-2020s righty-geezer days, that's all faded to an emotional zero; but I still can't feel any warmth towards the president who signed both the 1964 Civil Rights Act, which destroyed our freedom of association, and the 1965 Immigration Act, which destroyed our demographic stability. Still, I've ticked one more presidential birthplace off the list.
P.M. we hiked around the Pedernales Falls park. Enchanting scenery, lotsa bare rock
Government digits. Wednesday, March 29th. I indulged my keen-gardener wife this morning with a visit to Wildseeds Farm & Vineyard back in Fredericksburg.
I'm a horticultural dunce and have no worthwhile opinions in that area; but the lunch tray was as good as a lunch tray can be, and the wine excellent. And yes, my lady had fun among the flowers.
In the afternoon to Enchanted Rock, an outstanding illustration of our host's geological theory — a colossal bare granite dome decorated with … huge rocks. Good feeling of triumph at having hiked up there without mishap or aneurysm, sensational views.
Not having thought to look it up beforehand, I wanted to know how high the summit was. I scouted around for some kind of survey marker with a number on it.
Sure enough I found one. It was old and grimy, set into the rock at the summit. There was too much dirt and corrosion to read clearly what it said, but I could make out some digits.
I got down on my knees and brushed it clean as best I could with my hankie. Yep, there were digits.
U.S. Coast and Geodetic Survey Reference Mark
For information write to the Director, Washington D.C.
$250 fine or imprisonment for disturbing this mark.
Something to remember. Thursday, March 30th. To San Antonio. First stop of course the Alamo.
Definitely something to remember. We took a million pictures: the Mrs with Davy Crockett, me with re-enacters, etc. Having just recently read all 22 of the Richard Sharpe novels I had a head full of information about early-19th-century long guns. That made for an interesting and informative (on both sides, I hope) conversation with the re-enacters.
I did not know until today that several people who were in the Alamo when Santa Anna's troop assaulted it survived the horror and lived to tell the tale. There seems to be uncertainty about the exact number, with fourteen or fifteen most commonly stated. Almost all were women, children, or slaves.
When sated with the Alamo we signed up for a hop-on, hop-off bus tour of the city. Star feature: the San Antonio Museum of Art, which is all that a museum should be. Darn it, Texas is good at museums.
A city on a river. Friday, March 31st. One more day in San Antonio.
One of Mrs Derbyshire's friends in Long Island had enthused to her about the river walk, so we did some of that. Then, seeing a boat full of sightseers on the river, we made enquiries and took a boat ride ourselves.
Later we took in a botanical garden and a trip to the King William historic district. After sundown we strolled down the river walk again among the evening illuminations and took dinner in a riverside restaurant.
San Antonio? Full marks from the Derbs.
Invading the Capitol. Saturday, April 1st. With one more day in Texas we thought we'd check out Austin, the state capital.
For a while this seemed like a flop. We took another bus tour, but the tour guide didn't have much to show us. He seemed to nurse an obsession with live music.
"This district we're coming to is famous for live music — it's all over! … The street we're turning into here has great live music — the best! … Oh, and see this place here? Live music every night! …"
If there was anything worthy of note in Austin other than live music, we didn't get to hear about it. We bailed out of the bus tour and wandered aimlessly for half an hour. That's when I got the idea to check out the state Capitol.
At first this didn't look too promising either. The Capitol is a fine majestic building set among beautiful lawns and trees. On this day, however, it seemed to be having some serious work done, with construction fencing and scaffolding all around. I didn't think we'd be able to get in.
Taking a chance, I went up to the fencing at front. There was a uniformed cop at the other side of the fencing. Was there any way we could get in? I asked him.
"Sure! Just go along there …" He pointed to the left. "… See that gap? Through there and along a few yards, there's a door you can use."
We followed his directions and, sure enough, found ourselves inside the Capitol building.
There was a fair number of other sightseers in the building, strolling around looking at the decor, peering down the corridors, sometimes trying a door. A scattering of cops were there too, paying no apparent attention to the sightseers. We felt like January 6th protestors, although no-one was visibly protesting anything.
It's a splendid old building, immaculately kept. We wandered all over, up and down several of the grand stairways. We admired the portraits of Texas governors. And yes, we tried doors. Some of them opened, leading to finely-appointed common rooms. Others, notably the door to the governor's office, didn't.
There didn't seem to be any legislators in the building. Remembering what I'd learned from The Economist about the legislature only meeting every other year, I cornered a cop and asked him whether this was an on year or an off year. It's an on year, he assured me.
"But there don't seem to be any legislators here."
He laughed. "You kidding? They don't work weekends."
I was liking Texas more and more.
For state rights. My Texas-appreciation-ometer had not yet recorded its maximum reading, though.
After touring the Capitol building, we mooched around on the lawns out front. There was some kind of social event going on: dozens of people, mostly twenty-somethings, loud and tipsy and happy in their best suits and dresses. We thought it was a wedding, but when we asked they told us it was a college graduation ceremony, or the aftermath thereof. Early April seems like an odd sort of time to graduate college, but I guess Texans know their own business best.
Wandering down to the edges of the lawn we came across a grand monument — to the Confederacy! The topmost feature is a statue of Jeff Davis.
The legend on the plinth says:
FOR STATE RIGHTS
GUARANTEED UNDER THE CONSTITUTION
The people of the South, animated by the spirit of 1776, to preserve their rights, withdrew from the federal compact in 1861 …
And Texas joined them, against the protestations of Governor Sam Houston (who would have preferred that Texas revert to being an independent republic once again). And here was a handsome monument to the event, right on the lawns of the Texas state Capitol, in what everyone had told me was the bluest city in the state!
Hadn't the wokesters of Austin tried to vandalize this monument? I asked one of the graduation celebrants. He: "In Texas? They wouldn't dare."
Or perhaps they're just too busy listening to live music.
Is civilization better than barbarism? Here's a curious thing. I don't think it rises to the level of being a coincidence, but it's struck me as … curious.
As I related in last month's Diary, for reading matter on the plane ride to Texas I chose the 1901 novel Slave of the Huns by Hungarian author Géza Gárdonyi. The novel is enough of a classic to have its own Wikipedia page, and there's a good brief (150 words) summary of its content there.
A key inspiration for Gárdonyi was, as I noted, the embassy sent by the Eastern Roman Emperor in a.d. 448 to the court of Attila the Hun. Mingling with the Huns at Attila's camp, the embassy scribe Priscus was surprised when a Hun nobleman addressed him in Greek.
The nobleman, it turned out, had originally been a citizen of the Eastern, Greek-speaking, Roman Empire who had been captured and enslaved when the Huns sacked one of the Balkan cities. After doing good service for his Hunnish masters, however, they had freed him, and he had risen to wealth and power among them. Gibbon, in Decline and Fall of the Roman Empire, tells us that:
The freedman … exposed, in true and lively colours, the vices of a declining empire of which he had so long been the victim; the cruel absurdity of the Roman princes, unable to protect their subjects against the public enemy, unwilling to trust them with arms for their own defence; the intolerable weight of taxes, rendered still more oppressive by the intricate or arbitrary modes of collection; the obscurity of numerous and contradictory laws; the tedious and expensive forms of judicial proceedings; the partial administration of justice; and the universal corruption which increased the influence of the rich and aggravated the misfortunes of the poor. A sentiment of patriotic sympathy was at length revived in the breast of the fortunate exile; and he lamented, with a flood of tears, the guilt or weakness of those magistrates who had perverted the wisest and most salutary institutions.
I shall leave you to ponder in your own time the relevance of that list of imperial vices to the present-day U.S.A. The point I want to dwell on here is, that a man raised in a civilized state became by chance the subject of a barbarian prince, and settled in happily to his changed cicumstances. To this nameless Greek-turned-Hun, plainly civilization was not better than barbarism.
Well, that's a fun thought to kick around. The curious thing is, that visiting Texas opened my awareness to numerous similar cases from the 1860s and 1870s.
Browsing the books on sale at the museums and historical sites we visited, I kept noticing that the nonfiction shelves always included at least one book about the white children captured by raiding Indians in the Hill Country during those years: books like Herman Lehmann's Nine Years Among the Indians, Smith's and Hunter's The Boy Captives, and Scott Zesch's The Captured.
My interest piqued, I read Herman Lehmann's autobiographical account. Lehmann was captured at age not-quite-eleven by Apaches in a raid on his family's farm twenty miles north of Fredericksburg. That was May, 1870. He spent the next eight years living with the Indians. (And yes, that contradicts the title of his book, which came out in 1927. Presumably Lehmann or his publishers exaggerated a little. Believe it or not, authors and publishers have been known to do that.)
There's some grisly stuff in there. The Apaches really were barbarous when they chose to be. Here's an episode from Lehmann's career as a thoroughly acculturated Apache raider:
We came into a plain road and discovered a wagon in which there was a man and woman and three children. Before they were aware of our presence we had them surrounded, and it was but the work of a few seconds to kill and scalp the man, woman, and a little baby.
They scalped the baby? Ay ay ay!
The two other children we carried with us, a little girl about eight, and a boy about six years old.
The children proved to be a nuisance, though. They wouldn't eat and cried all the time. Possibly they were upset at having seen their parents and baby sibling murdered and scalped.
We could not slip up and steal anything for the noise they made, so …
So the Apaches killed them. They threw them high up in the air, two Indians per child, trampled the broken bodies by riding horses over them, then hung up the mangled remains for vultures to eat.
The astonishing thing is that, horrors like that notwithstanding, Lehmann, like the Greek in Attila's camp, came to prefer the barbarian way of life. Writing (or dictating) his memoir long afterwards, he spoke longingly of "the freedom of the Plains." By age sixteen he was a sort of assistant chief, leader of a small Indian band.
At that point, after six years with the Apaches, Lehmann killed one of their medicine men and had to flee for his life. His squaw urged him to go back to his own people, but Lehmann didn't want to. Instead he joined a Comanche band and spent two years with them.
And this was the rule, not the exception, among these child captives. Zesch's book tabulates nine of them, seven male and two female. Their ages at capture, females asterisked, were 13, 10*, 14, 8*, 8, 10, 11, 10, 8. Time in captivity: 12 years, 7 months*, 9 months, 6 months*, 5 years, 3 years, 8 years, 20 months, 3 months.
The ones longest in captivity were the most reluctant to return to the white world. When they were forced to, as the tribes were hustled off into reservations in the 1870s, they were usually miserable. (Neither of the females, by the way, recorded anything but kindness from their Indian captors.)
An extreme case was poor Temple Friend, captured at age eight, five years with the Comanches. Back in civilization, he hated school so much his father allowed him to quit. That didn't help. Temple went into a decline and died at age fifteen of an ailment no-one could identify. "A broken heart," is Zesch's diagnosis.
The answer to the title question here seems to be that if you are a teenage boy, robust, in good health, and not much interested in reading or intellectual pursuits, barbarism is better than civilization. Zesch:
The Southern Plains Indians and their captives lived in societies that imposed few rules and punishments. Young men learned how to behave by observing their elders, not by being taught a moral code. As long as the tribe had enough to eat and wasn't being threatened by enemies, camp life was relaxed and unfettered. The Indians gave their young free rein when it came to pleasures such as eating, drinking, sleeping, and gambling.
It doesn't sound bad at all.
All places, all times. Those child captives of the 1860s-70s Texas Hill Country were of course only a tiny fraction of white American settlers who made a home among the Indians.
Many did so volutarily: Sam Houston, for example, much earlier in the 19th century.
He ran away from home in 1809 and spent nearly three years living among the Cherokee in eastern Tennessee. Adopted by a clan led by Chief Oolooteka, Houston learned to speak the Cherokee language and adopted many of their customs; they gave him the Indian name Colonneh, or "the Raven."
There was some serious white-Indian mixing from the earliest days of settlement here, a lot of it — I'd guess most of it — voluntary. As I noted after visiting the battle site of Horseshoe Bend in Alabama twenty years ago:
I suppose that this battle is nowadays taught to our children as a victory by rapacious white savages over gentle, spiritual, peace-loving farming and weaving folk of the "Native American" persuasion. It is therefore with malicious glee that I report the following fact: After a century or so of frontier mingling and miscegenation, the Creeks numbered among their leaders men bearing names like Paddy Walsh, Josiah Francis, Peter McQueen and William Weatherford (a/k/a Chief Red Eagle).
And these cases themselves are only a tiny proportion of a phenomenon that we can safely presume to have been world-wide across thousands of years: the phenomenon of literate, civilized people settled somehow or other, voluntarily or not, among barbarians. I bet there were Bronze-Age kids with tales much like Herman Lehmann's: Hittites or Elamites or Achaeans snatched from their (relatively) civilized homes by barbarian raiders from the steppes or the Caucasus mountains.
For sure China wasn't spared. From the earliest days of Chinese civilization there were illiterate barbarian peoples just across the Empire's borders, sometimes surging through those borders and seizing imperial territory, occasionally taking the Empire itself.
Chinese commentators traditionally divided the barbarians on a simple compass-points model: Northern, Southern, Eastern, and Western, with a blithe indifference to fine ethnic distinctions. Mongolians, Turks, Siberians, Tibetans, Koreans, even steppe-Iranian tribes were in the barbarian mix at various times along with Central- and Southeast-Asian peoples whose precise lineage historians and ethnologists are still squabbling about.
The most famous case of a civilized Chinese person among barbarians is in the story of Wáng Zhāojūn in the first century B.C., which I've told in prose and verse on a "Readings" page. There must have been untold thousands of others, voluntary and otherwise.
I doubt they all came to believe that barbarism is better than civilization. The Central Asian steppe, never mind Siberia or Tibet, is a harsher environment than Texas Hill Country; and the lifestyle of all but the poorest Chinese people, even two thousand years ago, had more creature comforts than the American frontier in 1870.
This is all a dead letter nowadays, of course. There are precious few illiterate barbarian peoples left. (I know: I'm tempting readers to naughty quips about inner-city Baltimore and Minneapolis. Resist the temptation, please.)
If you want to give the barbarian lifestyle a try you could, I suppose, try to get in among the Sentinel Islanders, but they don't seem to be very hospitable.
So: civilization, barbarism, what do I think? I much prefer civilization, and I think I would have done so at any point in my life. The fact that some other sane, normal people have felt otherwise is none the less worth pondering.
Where are the Huns? Here again is that Greek chap among the Huns, grumbling about life in late-imperial Rome a couple of segments ago:
The intolerable weight of taxes, rendered still more oppressive by the intricate or arbitrary modes of collection …
And here is "Regime Critic" Josiah Lippincott at American Greatness, April 17th, in an article titled "The Tax Preparation Industry Deserves to Die":
This country is one giant sheep shearing operation. We are deep indeed into the looting phase of late-stage imperial decline.
Yes, it's tax-prep time. Phone calls from the accountant, emails to the accountant, visits to the accountant, … the damn business gets more convoluted every year. What the hell is a K-1?
We have, of course, been boiled like the proverbial frog: a new regulation here, a new deduction there, … year after year to the monstrous abomination that is today's tax code.
Why? Lippincott blames it on tax-prep software firms like Intuit buying up congresscritters. I'm sure that hasn't helped, but the complexification would have happened anyway. I was doing my taxes before tax-prep software came up, and it was already happening.
Mr Parkinson taught us 68 years ago that "Work expands so as to fill the time available for its completion." Some similar principle applies to tax regulations: they expand to exhaust the tolerance of citizens for having their free time taken away from them by force of law.
Didn't some presidential candidate a few election cycles ago propose a flat tax to replace the current monstrosity? Yes, there was such a candidate — more than one, more than once. So what happened to that idea? Same thing that happens to all sane and sensible political proposals nowadays if they give ordinary citizens an advantage against the ruling class. [Sound of toilet flushing.]
"Late-stage imperial decline"? Sure looks like it. Where are the Huns when we need them?
Will AI eat us? Much talk this month about AI. Is it an existential danger to our species? Some people very well-credentialed in the AI field itself think so, including a guy who's been working on it for fifty years.
I am not well-credentialed in the AI field and don't have an opinion on the matter. I do, though, have a mild vexation. Here's my mild vexation.
I studied math at University College, London, 1963-66. (No, I didn't drop out. Three years was standard for an undergraduate degree in Britain at that time, and for all I know still is.)
UCL had a good rich collection of discussion societies we could join. One of them was the Humanist Society, for agnostics and atheists. They published a periodical with the title Ethic in which students who cared to could air their opinions on matters related to humanism, however distantly.
I wasn't a member of Humanist Soc. but a classmate of mine had some kind of position with them. He was always canvassing the Math Department for contributions. Like any twentyish college student I had a head full of opinions, so one day I yielded to my classmate's canvassing and worked up an opinion piece a few hundred words long that was published in Ethic.
I can't remember the subject of that article. With one exception, I remember nothing of what I wrote. I seriously doubt there was anything much worth remembering. I was introverted, unworldly, and not terrifically smart — certainly no Frank Ramsey.
The one tiny thing I can remember is that I predicted AI+, the form of intelligence that doesn't just mimic the human variety but goes beyond it to superintelligence. I wrote something like: "In the near future, perhaps before the end of this century, we shall be sharing our planet with beings more intelligent than ourselves."
Of course I forget the context in which I wrote that, but I do remember writing it.
Was I displaying sensational powers of precognition? Hardly. I had spent my teen years reading almost nothing but science fiction, and the notion of superintelligent computers was already a theme therein.
Stanley Kubrick's movie 2001: A Space Odyssey with its intelligent on-board computer HAL came out in 1968; my contribution to Ethic was somewhere in 1964-1966.
All right, HAL wasn't superintelligent, but he was artificially intelligent. The concept of AI was already in the mid-1960s air, at any rate for sci-fi buffs. Superintelligence — "beings more intelligent than ourselves" — was a slight and obvious extrapolation.
So what am I vexed about? Well, that Ethic article was the first piece of any length I wrote for publication. I'd like to have it for my personal collection. The original was lost in my travels decades ago but I nurse the hope that a copy has survived somewhere.
University College, London still had a Humanist Society at least up to five years ago, when they they posted on Facebook that:
It is with sad hearts we hereby announce UCL Atheist, Secularist and Humanist Society to be dormant.
I've tried to get in touch with possible contacts online but nothing came back.
A librarian friend suggested I try the online catalog for the library at Harvard Divinity School. "They likely have atheist and humanist journals in their archives," she told me. I gave it a try and, yes! there it was.
The downward links took me nowhere, though, and the "Chat with a Librarian" facility left me with the impression that I would actually have to show up at the library in person and go through the archives, with small prospect of success even then.
Cambridge, Mass. is too far to go to salve a mild vexation. I think I'll wait until online library searches get more user-friendly. Someone must be working on that. Someone, or … some AI …
Math corner. I shall pass on a brainteaser this month, mainly from annoyance at not having solved last month's. How difficult can the durn thing be? I'm probably overthinking it.
On the matter of DEI in math, readers have been asking me to comment on the story that showed up at the Scientific American website, April 10th.
Two high school students have proved the Pythagorean theorem in a way that one early 20th-century mathematician thought was impossible: using trigonometry.
Calcea Johnson and Ne'Kiya Jackson, both at St. Mary's Academy in New Orleans, announced their achievement last month at an American Mathematical Society meeting.
The Pythagorean Theorem, I'm sure I don't need to tell you, concerns right-angled plane triangles. The square of the length of the longest side — the hypoteneuse — is equal to the sum of the squares of the lengths of the two shorter sides.
People have been cooking up proofs of the theorem for centuries. In 1927 Elisha Loomis, that "early 20th-century mathematician" Scientific American refers to, published a book containing 371 different proofs. Amazon will sell you a copy of Loomis' book for $1,413.54. (How on earth do they come up with these prices?) You can read the 1940 second edition, which is somewhat shorter, online for nothing at all.
Why, then, is one more proof noteworthy? Well, the Misses Johnson and Jackson use trigonometry in their proof. Loomis ruled that out of court because, he said, "all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem," so that a trigonometric proof must employ circular logic.
Johnson and Jackson have not so far presented their proof to us as a published paper, so it's hard to know what to make of it.
The best we have to go on is a YouTube clip at the MathTrain channel that tries to reconstruct the proof from still pictures taken at the AMS presentation. If that really is the Johnson and Jackson proof, it's pretty nifty for high-schoolers. Also, I think, valid, whatever Mr Loomis might have said.
The editor of the MathTrain video warns at the top of his Comments column:
Please be nice in the comments — all comments discrediting or disparaging the youngsters will be deleted.
Johnson and Jackson both being black, that implies that some commenters have suspected an anti-white swindle here. I wouldn't rule it out, given the command that anti-whiteness has over our culture, especially our schools, and the far-gone wokery of the American Mathematical Society. In fairness to these young ladies, though, we should suspend doubt until either (a) we see a valid proof published under their names in a peer-reviewed journal, or (b) a swindle is exposed.