The 2nd cheapest car in America, the Awesome Chevy Trax, a 3 cylinder SUV with zip

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I took a trip to Princeton this weekend and rented the cheapest car I could, a 2026 Chevy Trax, rented from Drivo -Ace for $25.47/day. Taxes and state fees added another $26.50 for a total of $52/day. Still not bad. I’d known that this car had a ridiculously small 3 cylinder engine, 1.2 liter. I thus expected the car to be tiny, and expected no power or acceleration. In reality, it was spacious and attractive, a compact SUV with decent size seats front and back, and a lot of trunk space, 54 cubic feet of storage with the back seat down. And it was peppy, not great, but not sluggish at all. Zero to 60 in ~ 8.7 seconds, respectable, far better than the 10 seconds or so for the Nissan Versa. Congratulations to GM’s Changwon, Korea division who makes the car, and to GM who approved the project. The gas milage was great 30 mpg in my combined hwy and city driving — real world driving with me at the wheel. With milage like that, you’d think this is a hybrid, but it isn’t, and it doesn’t need fancy gas, just good old 87 octane.

2026 Chevy Trax in blah – grey. Also available in colors.

I have to credit the turbocharger for the high mpg and zip, along with the light weight, about 3000 lbs depending on the trim — I didn’t think to check the trim or weight because I didn’t expect to write this blog until after I’d returned the car. Most US cars have gotten really heavy for no obvious reason, over 4000 lbs typically. To get similar acceleration, US cars generally go to a bigger engine often non-turbocharged. They often pair it with a variable transmission that tends to feel funky. The trax uses a standard transmission.

The price for this vehicle is $21,000 to $23, 500 depending on trim, slightly higher than the cheapest for new car in America. The 2026 Hyundai Venue, comes a base price of $20,550, before taxes, fees, and dealer incentives, but after these things, the price ends up the same as the Trax. I notice that the Venue is smaller though, with a bigger engine 1.6 liter (not turbocharged), and continuously variable transmission. Because of the continuous transmission, I expect that you can not manually shift – something I like to do; I know if I’m going to accelerate soon, or plan to just cruse while the computer does not. The Hyundai Venue gets similar mpg to the Trax, but does not come with lane following, at least at the lower trim. The basic Trax comes with OK lane following that’s active only when you go over a certain speed. There’s a small screen on the Trax, and real buttons, plus a manual shift adjuster that I was glad to see, but din’t use on my trip. I din’t find the Trax loud either, or rough riding despite the 3 cylinders, my guess is the smooth-ish ride is a result of some power balancing from the turbo, maybe. I’ve never driven a Venue, BTW.

The biggest cost of a car, typically, is depreciation, so it generally pays to buy used. And since I’m looking to buy a car, I looked into buying used Trax a 2026 or early with perhaps 20,000 miles, expecting to find a bargain now that the 2027s are out. As it turns out, a new 2026 is no cheaper than the new 2027. And a used 2025 with a few thousand miles costs just a few thousand less. This low depreciation suggests that a new Trax is a relatively good deal, though I’m still inclined to go used, e.g. a 2024 model for $15,500. My compliments to GM for delivering a fun, affordable, attractive car. Congrats.

Robert Buxbaum, November 4, 2026. My trip to Princeton was great, BTW. It’s a gorgeous town, and I got together with folks I like that I hadn’t seen in 45 years or so. We’re getting older, too.

It was better in Dubai before Iran attacked

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Dubai is a gorgeous, clean, modern city in the Emirates state of Abu Dhabi. As such it is Abu Dhabi’ as it were, Abu Dhabi’s gateway into being a modern Islamic Republic. All religions are respected in the city of Dubai, though not elsewhere in the Emirates. Alcohol and nightclubs are permitted, even encouraged, along with western books, movies, and music. There’s fast Internet, low taxes, free newspapers, and great plane connections via Emirates Air, along with premiere hotels, Ferraris, lots of them, and villas on peninsulas shaped like palm trees. Dubai is promoted with the phrase, “It’s better in Dubai,” Other promotions included having the Sex in the City movie filmed there and hosting the 2022 FIFA World Cup. And then Iran attacked with 5000 drones and missiles, hitting hotels, below and oil shipping. Three Emirates oil ships struck in the last week alone, leaving one crew member dead, 20 injured. Some rich people are fleeing, while others are staying. The airport is still open, as are the eyes of some rich folks in the US and Europe. It’s not so great in Dubai, but not that bad either. I don’t think Dubai is going away, and expect it to grow, slowly.

Photos from Dubai after an attack last week.

A similar pattern is playing out in Qatar’s western-style area of Doha. This is a section of the capital city that allows alcohol, and has fast internet, air-conditioning, western religion, movies, and books. I’d written, with wonder about Qatar, with its rich citizens and indentured, workers. I would have thought Doha would be safe from Iranian missiles since Qatar is Shia Mulsim, like Iran. It too is being hit, and not only in the western areas. Iran also destroyed Qatar’s largest natural gas and oil facilities, and their helium plant. The problem for Iran is that Qatar isn’t Shia Islamic enough to make them happy. Alternately, Qatar is rich, and Iran is jealous. Destruction is the answer, but the city endures, and I expect it will grow.

Saudi Arabia was in the process of building the beginnings of its own, modern gateway city, NEOM. It’s more conservative religiously than Dubai, but more technically advanced, envisioned with robots, solar energy, and zero cars. It’s in the early stages, but already attacked by Iran. The motivation, I expect, is to insure that the king doesn’t sign the Abraham accords with Israel and Abu Dhabi. The king has slowed construction, but not stopped, or stopped working with the US.

While these attacks are clearly a setback for these western-enclaves, I don’t expect Saudi Arabia, Qatar, and Abu Dhabi will shut them down. Dubai, Doha, and Neom are attractive, and seen as a necessary part of a post oil future. I suspect the attacks only bring resentment of Iran and its mullahs, plus a desire for increased spending on defense. The attacked countries may ally with each other, something unlikely before the attacks perhaps through expended Abraham accords, or via some other defense cooperation. I also imagine that the gateway cities will grow, allowing conservative Muslims to become more comfortable with Western ideas and behaviors.

I don’t expect Iran to change, nor copy these cities, nor do I expect their influence to weaken much. They have oil and a powerful ally, China, who depends on Iranian oil. China benefits by Iran fighting China’s western rivals, or it did before sanctions. I expect Iran to make more trouble for us and their neighbors, especially when Democrats win congress in a few months. I’m not totally sure why Democrats are sanguine about a nuclear Iran as negotiated by Obama, or seem OK with Iran bombing their neighbors for no good reason. Then again, I don’t quite understand “Queers for Palestine.” Most liberals are unhappy, and my guess is that the Iran leadership is too. Both are happy seeing Dubai and Doha bombed. Even with the bombs falling, I see these cities as a gate to a more peaceful, more tolerant Middle East.

Robert Buxbaum September 14, 2026

Hydrogen stripping by membrane extraction

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We occasionally make hydrogen stripper systems where the goal is to strip the hydrogen, and sometimes other gases from a mixed gas leaving everything else in place. We use metal membranes, coiled as shown below, if the gas to be removed is hydrogen. For other gases H2S, siloxanes, we use polymeric membranes. In either case, it’s not a product we advertise, but it’s worthy of a blog post.

A typical customer was a Cambridge researcher using gas chromatography with hydrogen as the carrier gas to separate a series of compounds. He now wanted to recover the separated compounds without the hydrogen.

A hydrogen extractor during manufacture.

To minimize the amount of membrane needed, I like to create a plug flow. If the flow is low, or there isn’t much hydrogen, we use a single coiled membrane. If the flow is high or there is a lot of hydrogen, we use several straight membranes in parallel, typically with baffles; we even have a patient for this. Some customers want to recover the hydrogen. If so, we extract it with a vacuum pump. Other customers just want the hydrogen gone. For these oxidation works well. The reaction H2 + 1/2 O2 –> H2O is essentially irreversible, and quite fast on a palladium surface.

A basic analysis is to assume that flux is proportional to hydrogen pressure in the gas, and calculate the hydrogen pressure as total pressure, P, times the fraction of hydrogen in the mixed gas, H/(H + inerts) .

Then, for any small amount of membrane area, dA, the small amount of hydrogen removed per second, dH equals the area, dA, times the permeance of the membrane, P, times P, times H/(H + inerts): dH = -PP (H/(H+Inerts)) dA.

We now rearrange the above: dH(H+Inerts)/H = -PP dA, and integrate:

∫ dH ( H+Inerts)/H = ∫(1 + inerts/H) dH = ∫ -PP dA. It’s found that

Hi – Hf , + inerts • ln (Hi)/(Hf) = PPA. Thus,

A = (Hi – Hf )/PP + (Inerts/PP) ln (Hi)/(Hf).

where A is the total area of membrane needed, inerts = the inert gas flow per second, and and Hi and Hf ate the hydrogen flow rates in the mix at the beginning and end of the extraction. Hi – Hf equals the total amount of hydrogen removed per second.

Note that the first term dominates in the beginning of the extraction process, or where there is a lot of hydrogen content, ln 1 =0), and the pressure of hydrogen is essentially P. The logarithmic term dominates in latter (final) stages, where the inert content is the majority and hydrogen content changes fast. In these later stages, very little hydrogen removal decreases the partial pressure by half, but a significant area of membrane is needed to remove every successive 50% since the hydrogen pressure is continuously decreasing by 50%.

You can derive similar equations for normal distillation, as used to make liquor or refine gasoline. You can also modify the above expression because permeation isn’t always proportional to pressure but to the difference of Pressure to some power, the 1/2 power, or so. I should also mention that, in membrane distillation and liquor it is important to continuously remove components that can build up in the still. For membrane- based hydrogen stripping, we do this using a back-pressure controller, or a flow controller. Thus, if you know you have a feed of 95% hydrogen, 5% desirable inerts, you want a to maintain a suitable back-pressure and you want a flow controller that lets mostly inerts out of the process with minimal H2.

She was only the moonshiner’s daughter, but he lover her still.

Robert Buxbaum, September 6, 2026.

Michigan’s only sad container port, Monroe

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Michigan has only one container-capable port, the port of Monroe, shown below. It’s small, connected to Lake Erie via a channel that’s 500 ft wide but only 21 feet deep. This is OK for handling Great Lakes, wide-bottom rafts and some shallow draft barges, but it is too shallow to handle ocean going vessels, or even larger Great Lakes ships. The only way Port Monroe takes containers is by pushing them onto rafts that are then drawn by tug, typically to Canada. The port has been neglected due to a belief, popular with Michigan Democrats, that fast rail traffic is the future success, while ports are the past. This is a popular belief in Europe, Japan, and China, but planes, cars, and ships make more sense in the US I think, in part because of regulations that prevent rail-carried mail. A nice thing about planes and ships is we don’t buy land, nor maintain tracks.

In term of price per pound mile, ships are vastly cheaper than any other system, especially big ships. But big ships need deep water ports, ideally ports with good road (or rail) connections, and good connections to the sea. Monroe has good road and rail connections, and Detroit does too. Detroit is an obvious city for a hub, centrally located on the narrows between the US and Canada, but Detroit has no port infrastructure currently, and shows no interest in adding any. That leaves Monroe, pretty bad, but better than nothing.

The port of Monroe, MI, is a bulker port, mostly and handles no ferries or cruise ships. Some few containers leave by barge. The shallow channel is part of the problem. Picture from the Detroit News.

I would like to see the Port Monroe improved, with cranes and more storage area, and with the channel deepened to handle seaway – max vessels of 27 feet draft so we could handle ocean- cargo business, and cruise ships — a ferry would be nice too. With an improved Michigan port, our exports would not have to be trucked to New York for shipment, and we would be able to import machinery and steel without it having to stop in New York, or Montreal. The lack of a decent port is why Detroit is the most expensive major US city in terms of the cost of international trade.

Monroe has the space for improvement, and they got a recent USDoT grant of $15 million to improve. It’s covered the building of a new dock, and is expect to provide for some berths for Coast Guard ships and tugs (needed). With the new dock, there’s rumors that Monroe could be a decent way to export windmills components and other items that are too large for bridges and unsuitable to major ships. We’ll see how things develop.

Robert Buxbaum August 26, 2026

Helium shortage and alternatives.

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I’ve come to discover that we have a major helium shortage. Helium is used in semiconductor manufacture, in MRIs, chemical chromatography, and low temperature refrigeration. I need it for leak checking, something we just started doing, a quality assurance test for our products and now a service. I’m not hit hard by the shortage, but some other industries are.

For gas chromatography, helium can be replaced by hydrogen, as discussed here. I was recently invited to a seminar on it. Hydrogen gives sharper peaks than helium, and slightly quicker residence times, but it has to be very pure. As it happens, We make hydrogen purifiers at REB Research, it’s our main product. These purifiers allow you to do chromatography, or brazing with low-cost, bottled hydrogen, and even with electrolysis hydrogen. The purifier is a one-time purchase, and then you can use low cost hydrogen. You will want to vent the output of the chromatograph; you use very little for chromatography, or brazing, ~50 cc/min typically, but without some venting, it can build up eventually.

Helium monument Amarillo

Some testers have begun using natural gas for leak checking, I notice. It’s not a sensitive, and you have to make sure you’re vented so there isn’t a flame. A safer alternative is forming gas: 5% hydrogen in nitrogen. The leak sensitivity is low, well below that with helium, but above smoke, or smelly compounds.

A major cause of the helium draught is Biden’s sale, with congressional approval, June 24, 2024 of the Federal Helium Reserve. This was a massive underground stockpile of helium, 32 billion cubic feet, that filled a set of caverns that span three states. We’d managed the stockpile for years, with a monument too, see right, but Biden sold the entire reserve including 425 miles of piping, purifiers, and caverns to Messer LLC, a German company owned in China. See an NBC article about the sale, here. The price was $460 million, less than 1¢ per cubic foot of He, with the pipes, storage area, and purification facility thrown in for free. Messer LLC quickly sold off the helium to recoup their purchase, and now it’s gone.

The sale of the reserve would not have been so bad, but in March, the Iranian Revolutionary guard bombed Qatar’s Ras Laffan natural gas and helium plant, the world’s largest helium plant, doubling the spot price of helium, see. AP article here. Why does Iran, a Shia-Islamic republic, bomb plants in Qatar, another Shia Islamic republic? it’s a form of economic war. Russia is another major gas and helium producer, and I expect Ukraine will bomb Russia’s plants too. If you use helium for chromatography, you might want to switch to hydrogen, and invest in a purifier from REB Research.

Robert Buxbaum, July 31, 2026

The economic case for Alberta separatism

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Canada is poor relative to the US as I discussed in a previous post. Though they have lots of available land, home prices are very high, much higher than in the US because of regulations and fees. Similarly, it is hard to start a business in Canada, and salaries are lower. Again the major problems appear to be regulations and taxes. The Western Provences, Alberta, British Columbia, and Saskatchewan complain regularly that their interests are being ignored. So, earlier this year, Alberta’s legislature approved a referendum for separation from Canada. Independence! In part the reason is that Albertans find they are financing the rest of Canada. Independence is something that Quebec attempted Canada some 30 years ago. There the “exit” vote lost by a mere 0.6%, and the argument to stay was mostly economic: Quebec needed Canada’s money. Alberta does not. It funds Quebec and every other Provence of Canada.

Trump praised Alberta’s decision to Canada’s dismay. Some Albertans want to want to join the US, a fulfillment of James Knox Polk’s vision. It may not be legal, though. Alberta judge, Shaina Leonard declared Alberta separation unconstitutional as “secession from Canada will have an impact” on native groups. Why this is different from Quebec, I don’t know. The Alberta legislature is now preparing for a referendum on whether Alberta should “commence the legal process … to hold a binding provincial referendum”. This is a referendum on whether to have a referendum.

Currently Alberta is the only Provence that puts in more to the federal government than it takes out, and the disparity is huge. The Provence of Alberta has more oil than most oil countries, more that Iraq, Kuwait, or Norway, for example. It currently produces ~4.1 million barrels of oil per day. Valued at the current world rate of $100/bbl, that’s about $150 billion per year. If you divide by 5 million Albertans, the oil revenue alone is ~$30,000 per man woman or child, or $100,000 per voter. To this you can add bountiful natural gas, other minerals, and a strong industry, and you see a lot of info e that federal Canada “redistributes” east. Because of Canada’s parliamentary system Alberta has virtually no rights. Each vote is counted equally, and almost every one in Canada believes Albertans should not keep their bonanza. The central Canadian government provides programs, it’s true, healthcare with long waits and medical-assisted suicide. They also policed face masks for truckers, taking away the truckers savings and trucks if Canada found the mandates were being ignored.

Canada’s liberal politicians blast Albertans for being racist and stingy. They spend widely and run a deficit, about 15% of tax income despite the money taken from stingy Alberta. And despite this spending, managed to put Canada into recession, the only G7 country in recession, currently. Canadians are poorer than US citizens, with lower wages, and higher home prices, and Canada looking to China as an ally. I would suggest, instead, fewer regulations and less spending as Albertans want. If Alberta leaves, the rest of Canada will be bankrupt.

Current odds are against Albertans’ separation from Canada, but Quebec almost left, and the economic argument for leaving was non existent. Quebec was taking money from the rest of Canada when the Quebec separation vote came 30 years ago, and even so non-secession won by a mere 0.6%. Economically Alberta is more than self sufficient, and culturally Alberta is so like the US that it could join us easily, or relatively so. A vote for separation, if it were allowed to take place, could easily result in separation.

Robert Buxbaum, July 23, 2026

Cruise the Great Lakes or the St Lawrence

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Pearl of the Seas, one of several Great Lakes cruise companies.

The Great Lakes system presents a growing summer cruise opportunity, relatively untraveled, but so far relatively expensive. There’s great scenery, including Niagara Falls, Mackinac, Painted Rocks, Chicago, and the Soo Locks: interesting sites, and the options are growing. As a cheaper alternative, as interesting, IMHO, is the lower Saint Lawrence seaway, including Quebec City, Montreal, Portland and Prince Edward Island, leaving from Boston, NYC, or Philadelphia.

The 2026 Great Lakes season includes seven cruise lanes and 10 largish ships and 7 cruise lines, up from six lines and eight ships in 2025. Major lines include Viking, Pearl Seas, Ponant, and American Cruise Lines who provide over 800 port visits in 2026, up from 700 in 2025. The largest of the Great Lakes ships are smaller than those for Alaska and the Caribbean cruises, but more luxurious. On the St Lawrence, the ships are bigger, the prices lower, and you get more or less what you’d expect on an Alaska cruise, with an open bar, typically.

Viking Oceanis

The cost differences are more than I would expect, but coming down, I expect. Currently, a nine-day Great Lakes cruise from a Viking Oceanis cruise from Cleveland costs a minimum of $8,675 per person. The 14-day version in the map above costs $14,270+. Details: https://l.cleveland.com/g2f86w. By comparison, a Boston to Quebec City cruise costs about $1200 per person, similar to a one week European or Alaska tour, on average.

I put the blame for the high cost of Great Lakes cruising on the size limits of the locks on the St Lawrence canal and on the Niagara Falls, Welland Canal: 740 feet (225.5 m) in length, 78 feet (23.8 m) in width (beam), and 26.5 feet (8.08 m) of depth (draft). The largest cruise ship now operating on the Great Lakes, the Viking Oceanis, show above measures 665 feet (205 meters) length, by 77 feet (23.5 meters) width, by 19 feet, 8″ draft, barely smaller than the canal. It boasts a capacity for 378 guests and 256 crew members. By contrast, the Norwegian Jewel, a popular ship for the New York and Boston to Quebec City route holds 2,870 passenger and a crew of about 1,069. This explains some of the price difference but not all. The rest, perhaps is competition. A standard European cruise ship can be rerouted to the St. Lawrence, but not to the Great Lakes.

Aerial Norwegian Jewel Norwegian Jewel – Norwegian Cruise Line

If one wanted to run a large cruise ship on the Great Lakes, it would have to be built there. It could be done, but isn’t currently. We make very large ore carriers that operate entirely on the Great Lakes, so the capacity exists. The other alternative would be to expand one of the canals. IF this were done, there would be a lot more competition and ships, and we could be served by much larger container ships as well. An unfortunate aspect of the small lock size of the Welland and upper St. Lawrence is that currently, only very small container ships can enter or leave the Great Lakes. The result is that cargo prices are much higher than for any of the US coastal ports or the Mississippi. Both the US and Canada would benefit from a larger connection to the sea.

Robert Buxbaum, July 7, 2026

Our new helium leak detector at REB Research

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Our helium leak detector being calibrated with a standardized He leak

We just got a new helium leak detector, a Varian 938, about 15 years old, shown at right. Our old detector, a Veeco, had not worked well for years, and then we could no longer get it fixed. For a while we outsourced our leak checking, and on rush jobs it didn’t always get done. Now we can check everything in-house at much better sensitivity than before.

It used to be that helium leak checking services were common locally, in Detroit, but we lost a lot of manufacturing and our quality checkers left too, for Ohio, then abroad. Companies that used to do He leak checking switched to fluorescent die checking, which isn’t as good, and then stopped checking altogether. But small cracks at welds are not uncommon, particularly with laser welding of stainless steel. They form as the metal cools, and are too small to see. They show up in helium leakage through the micro-cracks, and if not detected and corrected, these cracks will grow and destroy a pipe or system over time. For my products, hydrogen purifiers, the micro cracks affect hydrogen purity. Any crack that helium can get through will allow impurities to diffuse into our output hydrogen. Because of the problems of micro-leaks and durability, it used to be obligatory in the nuclear navy to helium leak check every part.

The picture above was taken while calibrating our new helium leak detector. I’d bought some calibrated sources, in this case, one certified to bleed at 3.6×10-8 cc/sec, which is equivalent to 2.1 x 10-9 slpm. If a leak of that rate would appear in one of my 1 slpm purifiers, that would imply a delivered hydrogen purity of 99.999998%, and that’s generally the purity limit I certify. This is also a common leak standard in the industry, but my new detector is better than that standard. We can test down to 0.5×10-9cc/s . A 1 slpm purifier with that leak rate will deliver hydrogen at 99.9999999+ purity, more than nine-nines, as they say. On the calibration day, above, I tested three new purifiers, and found they were all perfect to the higher standard, ≤ 0.5×10-9cc/s. I’m so happy with this, I thinking of going into free-lance, helium leak checking. There’s no one local who does it.

In case you want to know, I bought this detector on eBay but found it didn’t work. I tried cleaning key parts, replaced the pump oil, nothing helped. Then bought another leak detector, with a spare filament, but without a pump or a mass-spec unit. It included the standard leaks that I now use to calibrate. That one didn’t work in the same way that the original one didn’t; we never reached pressure, and the filament never came on. Eventually, I came to believe that the vacuum pump was a fault. I sent it to a great pump repair shop in Greentown, PA, and discovered they also repair helium leak detectors. I hope to visit them. They replaced the pump and spiffed up my detector at a decent prices ($1500 as I recall). Now, it works like a charm.

Robert Buxbaum, July 1, 2026

The great NO KINGS rally speaker, Marcus Cicero

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A proven way to remove a leader is to accuse him of trying to become a king. The Democrats keep trying this, but they lack a great orator to make the case in a way that gets the guy out. The classic speaker for this, someone most everyone in college read, was Marcus Tullius Cicero. Take, for example, his speech against Catiline, his fellow senator, that got Catiline removed, and then murdered. I’ve copied a bit below, in translation from Latin.

WHEN, O Catiline, do you mean to cease abusing our patience? How long is that madness of yours still to mock us? When is there to be an end of that unbridled audacity of yours, swaggering about as it does now? Do not the nightly guards placed on the Palatine Hill—do not the watches posted throughout the city—does not the alarm of the people, and the union of all good men—does not the precaution taken of assembling the senate in this most defensible place—do not the looks and countenances of this venerable body here present, have any effect upon you?
Do you not feel that your plans are detected? Do you not see that your conspiracy is already arrested and rendered powerless by the knowledge which every one here possesses of it? What is there that you did last night, what the night before—where is it that you were—who was there that you summoned to meet you—what design was there which was adopted by you, with which you think that any one of us is unacquainted?
Shame on the age and on its principles! The senate is aware of these things; the consul sees them; and yet this man lives. Lives! aye, he comes even into the senate. He takes a part in the public deliberations; he is watching and marking down and checking off for slaughter every individual among us. And we, gallant men that we are, think that we are doing our duty to the republic if we keep out of the way of his frenzied attacks.
You ought, O Catiline, long ago to have been led to execution by command of the consul. That destruction which you have been long plotting against us ought to have already fallen on your own head……
……. speech, source.

It’s strong stuff, far better than the democrats have yet presented, with lines of innuendo and attack that have been reused many times since. Cicero claims to understand the thoughts of Catiline as he sits in the senate. He accuses Catiline, not only of wanting to be king, but of wanting their deaths: “He is watching and marking down and checking off for slaughter every individual among us.” Catiline was exiled, and then, with Cicero’s encouragement executed without trial. Julius Caesar argued for Catiline, claiming he should be imprisoned, but no more. So Caesar was exiled to fight the Britains and Gauls.

In time, Cicero and his friends came to believe that Caesar too dreams of kingship too, He had become too successful and uncontrollable. He needed to be re-educated, according to Cicero (below).

“Men who become uncontrollable and overconfident as a result of success need to be put through a round of training in reason until they recognize the fragility of human affairs and the uncertainty of fortune.”

Marcus Cato’s assessment was even stronger, in the senate he, argued that Caesar’s fight with the Gauls was illegal and immoral, and that he should be stripped of power. Democrats argue this today regarding the war in Venezuela and Iran. I think they are wrong, but the argument is the same, that we/I are the rightful power, and I/we know the evil plans in Caesar’s mind. The senate agreed to strip Caesar of power, but use deception getting it. Caesar, knowing how this would likely turn out, returned to Rome, fought Cicero’s and Cato’s friends, and pardoned Cicero 45 BCE. Cato committed suicide. About 1 year later, though, on the 15th of March, a group of Caesars so-called friends stabbed him to death in the senate, as recounted here. Cicero was killed in the bloody civil war that followed. What goes around comes around. And a dictatorship ensued.

I’m not convinced that Cicero’s attacks were legitimate, nor the Democrats’ attacks. In both cases, the speakers claim to know the evil thoughts in the mind of the leader. In both cases, the proposed solution was leadership by group consensus — leadership by a group who could not agree on anything beyond their own unhappiness.

Legitimate or not, attacks by a good orator worked. They caused the death of Catiline and Caesar, and many others over the years. Generally, they don’t benefit the great speaker, long term, nor result in freedom. I’m reminded of Trotsky, here. The lesson for those pushing to depose Trump is you need a really good speaker, and you could get a leader who’s worse. Your speaker needs to convince the audiences that he knows what his enemy is thinking. So far, no Democrat has managed to convince an audience of this, or even vaguely that he knows what Trump will do next.

Robert Buxbaum, June 24, 2026

Finding whole number Pythagorean triangles, etc.

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One of the coolest old bits of math is the Pythagorean theorem. For any right triangle, the sum of the squares of the length of the legs equals the square of the length of the hypotenuse. That is a2 + b2 = c2.

The formula wasn’t invented by Pythagorus, but was known as early as 1800 BCE, in Babylon. What Pythagorus adds is the focus on whole number solutions, like 3, 4, 5. These are called Pythagorean triples, something important to Pythagorus because he believed that the essence of the world was whole numbers. It’s an approach that also is found with Democritus and Epicurus. Democracy, for example being, the idea that the whole of the national sprit is based on the will of unit individuals rather than the will of a single mystic who alone sees the political soul.

There is a formula for finding Pythagorean triples that was discovered, or first presented, by Euclid. Pick any two positive whole numbers, m and n where m>n. For any value of m and n he showed that there is a pythagorean triple where the hypotenuse equals m2+n2 and where one side equals m2-n2 and the other side is 2mn. For example, if you pick m=3 and n=2, you’ll find there is a triple with hypotenuse 13, and with legs 5 and 12. It’s a very cool formula, I think. Like pythagorus, I find a certain magic in Pythagorean triples.

The proof of this method is to note that (m2-n2)2 = m4 + n4 – 2m22n2. Now note that this is very similar to the formula for (m2+n2)2 = m4 + n4 + 2m2n2. We see that (m2+n2)2 = (m2-n2)2 + 4m2n2. These are all whole numbers since m and n are whole numbers, and they have the form for a pythagorus triple. This formula gives you all the “fundamental “primitive” triples, plus some others. Once you have these, you can find the remaining ripples by multiplying these by 2, 3, 4, etc.

There is another nice formula for whole-number triangles in 3 space, useful for boat building and the like — if you go to the shop and ask for someone to cut wood, they generally want to cut to whole numbers, usually whole inches, 5″ by 12″. With pythagorean triples, there is a better chance that your pieces will fit together. (I was designing and building boats a few posts back) Maybe I’ll go into it in another post.

There are 16 primitive Pythagorean triples of numbers up to 100.

(3, 4, 5)(5, 12, 13)(8, 15, 17)(7, 24, 25)
(20, 21, 29)(12, 35, 37)(9, 40, 41)(28, 45, 53)
(11, 60, 61)(16, 63, 65)(33, 56, 65)(48, 55, 73)
(13, 84, 85)(36, 77, 85)(39, 80, 89)(65, 72, 97)

Robert Buxbaum, June 1, 2026