Is this question reasonable?


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This is a sample question for middle school math, published by the International Baccalaureate (IB) program. I found it here.

Here is a graph I made of this equation, using Desmos:

Looking at this graph, you see that after about 10 minutes, there are 11 cars per minute going through the intersection – and that’s the most cars. After about 25 minutes, there are zero cars going through the intersection, and after that, there is a negative number of cars (!!!).

I don’t think this equation models anything having to do with any intersection I’ve ever visited. Instead, I think that any intersection controlled by a traffic light is going to be more periodic, that is to say, something like some mix of sine or cosine functions — obviously not middle school material.

Why there are no space-faring civilizations, and never will be

Very persuasive article explains why space travel is impossible. The main reason is gravity. Written by Allan Milne Lees; I found it on Medium.

Allan Milne Lees5 days ago·7 min read


Image credit: Air & Space Magazine

Despite the populist hype of billionaire Sci-Fi fanboys and a perpetual stream of Hollywood entertainments to the contrary, humans will never explore the galaxy in person. In fact, we won’t even explore our own solar system up close and personal. This is not merely because robotic missions can do the job 1,000% better for 1/1000th the cost. It’s because of two fundamental biological reasons.

The first is gravity. Everything about our bodies is evolved to function under a gravitational acceleration at sea level of approximately 9.8 meters per second squared (9.8m²). Our hearts pump blood up to our heads, fighting gravity every centimeter of the way. Our muscles and bones are as strong as they are because every part of our bodies is fighting gravity every moment of our lives. Our sense of balance, which orients us spatially, depends on gravity being constant in one direction only: straight down.

Without gravity, very bad things happen: the heart pumps too much blood to the head and too little to the lower extremities, leading to ocular distortions, crushing headaches, and nausea as the inner ear loses all sense of up and down. Our bones and muscles atrophy dramatically, even when hours each day are dedicated to exercises specifically designed with the intention of slowing down this decay. Put simply, our bodies are incapable of handling microgravity and despite the pictures of smiling astronauts merrily enjoying microgravity on the ISS, the harsh reality is that every single one of those astronauts pays a price very few of us would wish to incur.

The Sci-Fi fanboy response to this fundamental problem is either (a) to ignore it entirely, as per Musk and Bezos, or (b) claim that artificial gravity is the answer.

As Musk and Bezos are ignoring the problem we can likewise ignore them. So what about artificial gravity?

There are only two ways to create artificial gravity. The first is called “constant-g” which means that we accelerate our hypothetical space ship at a constant 9.8m² for the first half of the trip and then flip it around and decelerate it at a constant 9.8m² for the second half of the trip. Einstein’s insight that over areas too small to experience tidal effects such acceleration would be indistinguishable from regular gravity means that in theory Earth-style gravity could be induced in such a manner. Better yet, because the acceleration is constant, relativistic speeds will eventually be attained. In just 12 years (in the reference frame of the spacecraft) we could travel across our Milky Way galaxy. In a single human lifetime (in the reference frame of the spacecraft), under constant acceleration, we could reach the edge of the universe that’s observable from Earth. An Earth upon which, in that frame of reference, billions of years would have passed.

So with constant acceleration we get a “twofer.” Earth-identical gravity and the ability to traverse vast distances within a human lifetime. Problem solved!

Except that there is no way, theoretical or otherwise, to achieve constant acceleration of this magnitude. No propulsion mechanism, theoretical or otherwise, can overcome the problem of mass. In order to power the continual acceleration, our imaginary space ship is constrained by Newton’s observation that any action in a vacuum requires an equal and opposite reaction. In other words, to accelerate a mass of X by some amount of velocity we will need to discharge an equivalent amount of energy in the opposite direction. And that energy can only come from fuel. Which adds to the mass of our space ship. So now we need to expend more energy, which means we need more fuel, which means we’re now carrying even more mass, which means we need to expend even more energy, which means…

In other words, even with some imaginary technology that could convert matter into energy with 100% efficiency, there’s simply no way to get to 9.8m² constant acceleration for any meaningful amount of time. Sure, we can talk about things like an Alcubierre drive but then we’re just as entitled to say that Hogwarts will invent the Spaciamus drive to solve our problem instead. In other words, running off to hide inside imaginary “solutions” is no solution at all.

If constant acceleration can’t provide artificial gravity, what about centrifugal force? We all remember the rotating space station in 2001 A Space Odyssey and everyone knows that this was the only Sci-Fi movie ever to have utilized a science-based series of technologies. Plus, it’s easy to find on the Internet lots of schemes to create artificial gravity in this way, from tethering ships together and spinning them around a central axis to building enormous hollow rotating cylinders on the inside of which humans will experience Earth-like gravity. So, problem solved!

Except the movies and the Sci-Fi books mislead us, as is the way of popular entertainments.

First, the good news: if a person stood perfectly still and did not move in any way whatsoever, then centrifugal force could seem to mimic Earth-style gravity. Unfortunately, here’s the bad news: if they made any movement whatsoever, they would instantly be overcome by nausea and be disoriented.

Why is this? Imagine throwing a ball up into the air here on Earth. If you throw it straight up, it will come straight down, pulled by gravity toward the center of the Earth we’re standing on. But under conditions of “gravity” induced by centrifugal force, a ball thrown straight up will arc and fall away from the person who threw it because unlike here on Earth there’s a second force acting on the ball: centripetal force. As our inner ear orients us by means of reference to the constant downward force of gravity, this means that any movement at all — even something as minor as turning one’s head — would result in signals from the inner ear (responding to the centripetal force) jarring dramatically with the signals from our eyes. At best this would lead to our hypothetical human vomiting in a majestic arc; at worst it could render them incapable of any controlled movement whatsoever.

The diagrams below show the difference between gravity (or constant acceleration at 9.8m²) and a rotating object. On Earth there’s only one force acting on us: gravity. On our imaginary rotating artificial gravity environment there are two forces: centrifugal, and centripetal. And that makes all the difference in the world.

Perhaps this is why Bezos prefers to ignore the problem; it can’t be solved just by throwing money at it. As for Musk, he makes people with ADHD look like paragons of sustained concentration so he probably doesn’t even know the problem exists. But even if you don’t know a brick wall exists, it still kills you if you slam into it at 1,000 kilometers per hour.

Gravity, therefore, is one reason why human beings will never be a space-faring species. It’s also the reason why it’s highly unlikely any other species capable of developing suitable technologies would ever become space-faring either. All organisms are highly adapted to the environments in which they evolve and it is extremely difficult to sustain organisms outside of their natural environments for any significant period of time. Add it the problems of solar radiation, the deleterious effects of microgravity, and everything else associated with space travel and it’s apparent that Sci-Fi fanboy dreams are a very poor guide to the future.

There is a second major reason why we humans will never be a space-faring species: psychology.

Our brains are as much the result of selection pressures as our bodies. Like our bodies, our brains are highly adapted to life on Earth. As a primate group species adapted to foraging, we’re not well-suited to being cooped up in tiny cages. We become obese and we develop all manner of mental problems. Without access to natural cues like water and grass and trees, we become stressed. When forced to interact with the same small group of people for years without respite, we become irrational and angry, or conversely withdrawn and depressed. Worse still, our emotional hardwiring makes us competitive even when cooperation is the optimal strategy, and our intellectual limitations lead us to acquiring and then strongly defending irrational and harmful beliefs.

Imagine, therefore, a space ship upon which 200 hapless humans attempt to exist for years or even decades. Instead of looking to Star Trek as our inspiration, a more probable vision is depicted in One Flew Over The Cuckoo’s Nest or perhaps the concluding episodes of some trash reality TV show.

It is difficult to imagine any species capable of making spacecraft not having equivalent psychological limitations, albeit likely somewhat different from those that control our own behaviors.

There are many other reasons why humans will never spread across the galaxy, but these two should suffice to prove the contention. This does not mean, however, that there won’t be money to be made in enabling space tourism. A few days in microgravity, ensconced in a modestly comfortable environment with a small number of others, could be a very congenial way for the wealthy to break up the monotony of holidaying in the Hamptons or on a private island in the Bahamas. Sheltered in low orbit by the Earth’s magnetic field, the dangers of solar radiation are reduced to a perfectly acceptable level and likely no worse than a dozen trips in a private jet. Microgravity sex will no doubt become this century’s equivalent of the Mile High Club that was so popular among the early jet-setters of the 1960s and 1970s.

But beyond a few amusing days spent orbiting the Earth while watching one’s champagne bubble around one’s head, and after the inevitable disaster of Mars Colony One, we will accept the fact that robotic missions are the real future. And then we will expand our knowledge of the universe exponentially instead of wasting hundreds of billions of dollars on futile dead-end fanboy dreams.

Final Silvering Results, with Angel Guard

A few days ago, we silvered an 8” diameter 43” FL mirror that had previously been aluminized, and applied the Angel Guard coating.

We did a Ronchi test and some Foucault-Couder knife edge tests before stripping the aluminum and after the silver was applied.

To my amazement, we found that the mirror’s figure was about the same in both cases. How that works, especially how the Angel Guard coating is laid down so even and smooth over the entire mirror, is beyond me. But it DOES work.

Prior figure (aluminized mirror), seen with Ronchi grating of 100 lines per inch:…

Final figure (silvered mirror), with same Ronchi grating:…

This is a video of us washing off the Angel Guard coating.

Here is a video of the finished mirror after drying. Notice that the very edge of this mirror did not take the silver coating, but the area uncoated is probably on the order of one or two percent of the total area.

Silvering a Mirror (vs Aluminizing)

Ive been doing the aluminization process for telescope mirrors at the NCA ATM workshop with a 55-year-old military surplus aluminizer at a DC rec center for about 20 years. (I’ve had a lot of help!!) This involves high vacuum, a noisy pump, voltage both very high and very low, and quite a lot of time.

Today, I had the opportunity to silver a random piece of glass, in my driveway, with the aid of another longtime ATMer and some chemicals from Angel Gilding. I had seen this demonstrated at Stellafane by Howard Banich and Peter Pekurar in 2019.

Doing it myself was quite eye-opening.

Almost finished, except for the spray-on Angel-Guard, which we didn’t have

Here’s what I wrote on FB:

Success with our first attempt at silvering a piece of glass under a tent canopy, and then stripping off the silver quickly and easily with PCB etchant (FeCl?).

I’ve aluminized many mirrors with the NCA’s vacuum chamber, using a modified version of John Strong’s method from the 1930s.

I must admit that this method was faster, easier, quieter, and much more low-tech, compared to depositing aluminum. In the latter case, sometimes you have to wait an hour or more for the dual-stage vacuum chamber (the primary pump is VERY noisy!!) to finish all the preparatory steps and pump down low enough that a hot atom of gaseous aluminum can travel two or three feet before striking any other remaining air molecule! (That is one hell of a vacuum!)

With the silvering process, you can do any size mirror you can fit on your cleaning jig — and you can make it out of pieces of scrap wood, a few nylon chair legs, two old hinges, and some screws! !

With our NCA-ATM-CCCC aluminizer, we are limited to 12.5” max. I’m currently working on a 16.5” thin Pyrex mirror; the price I’m quoted for aluminizing it is about $600 at Majestic Coatings, which is about three times what I paid for the blank!! And that doesn’t even include shipping!

Today Alan T and I tested the silvering process in my driveway using the screen tent canopy that we use at the ATM workshop to stop dust particles from landing on mirrors that are being polished. (After getting permission to go retrieve the canopy from the Covid-closed rec center, we immediately went into the parking lot to hose off a decade of dust!!)

We unfortunately do not have Angel-Guard overcoating on hand. It should arrive Wednesday. As most folks know, bare silver, unlike bare aluminum, tarnishes very quickly (in weeks or months) when exposed to ordinary air, whereas a I have seen many bare Aluminum layers last a decade. This overcoating is said to extend the life of the coating to about a year, but obviously conditions will vary.

We used a 6″ float glass mirror blank to try out the process today – not an actual, parabolized mirror.

How does it work?

This is basically a five step process:

1. Get prepared and mix the tinning solution afresh;

2. Clean off the mirror properly with precipitated CaCO3 and/or Alconox; rinse;

3. Sensitize the mirror with an invisible layer of tin (Sn); rinse;

4. Spray on the silver solution and its reducer at the exact same time with two separate brand-new one-pint hand squirt bottles, until fully silvered & shiny; rinse;

5. Spray on the Angel-Guard overcoat; rinse; dry.

The amount of chemicals used is minimal. The nastiest stuff was the reducer. I’m glad we did this outside.

Btw: a number of people have bench-tested mirrors before and after this process. Some report no change in figure; somebody I trust, who has a Zygo interferometer, says there is a little degradation, but not much: a mirror that was 1/10 lambda (excellent) might go to 1/4 lambda, which is certainly usable for a big Dob, iirc.

And it’s cheap! And fast! And easy! And quiet!

We were able to fully and completely strip the brand new silver off with the PCB etchant in under 3 minutes. Aluminum takes much longer.

Thanks to: Léon Foucault; Steinheil; Howard Banich; Peter Pekurar; Angel Gilding; Alan Tarica; and my wife, Gail.

One spray for the silver solution, one spray for its reducer (chemistry)
It’s the tent canopy that’s bent, I think. But this is not a telescope mirror; just a 6 inch disk of float glass.
That’s Alan
I made the jig to hold the mirrors out of some scrap two-by-fours, some screws and reinforcement plates; two old hinges; some nylon chair feet; some 1/2″ PVC pipe; and some 1/2″ PVC end caps.
The tent-screen canopy needed staking and tying down to prevent it flying off. Putting up the frame took three people (me, Alan, and my wife, Gail).

A mental math trick

Can you do 994 times 997 in your head?

‘Shruti” will explain how to do it, but she doesn’t explain why it works:

Personally, I had never come across this one in my over 40 years of teaching math. She is correct, it does work, but WHY?

I’ll try to explain.

994 is 6 away from 1000, or 10^3 minus 6.

And 997 is 3 away from 1000, or 10^3 minus 3.

If we multiply (10^3 – 6) by (10^3 – 3), we get 10^6 – (6+3)*10^3 + 6*3

So as she explains, we count down from a million by nine thousand, and then we add on 18.

or, if we take away any small number from a thousand, say, a, and subtract any other small number b from a thousand, you have a situation like this if you use the area model for multiplication:

Clay Davies’ Links for Telescope Makers


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I am copying and pasting Clay Davies’ recent article published on a Facebook page for amateur telescope makers, where he gives links to extremely useful sources as well as commentary. I think he did a great job, and want to make this available to more people.

================================= here goes! ================================================

Amateur Telescope Making Resources & Fast Commercial Newtonian Telescopes

  • Observer’s Handbook, Royal Astronomical Society of Canada. Every amateur astronomer should have at least one copy of this book. Every “newby” should read it cover to cover. Old hands should keep it as a reference. Avid astronomers get it every year, because it’s updated annually.
  • How to Make a Telescope, Jean Texereau. A classic book by a superb optician. The author taught many people how to make their own telescopes, including grinding, polishing and figuring their own mirrors. This book offers unique and practical telescope and mount designs I have never seen anywhere else.
  • The Dobsonian Telescope, David Kriege & Richard Berry. Want to knock off an Obsession telescope? Here is your bible, written by the creators of Obsession Telescopes. Here you will find well thought out and time proven designs for truss Dobsonian telescopes from 12.5” to 25” and more. If you are handy, if you use one of these designs and follow step-by-step instructions, you can build a fine truss dobsonian. But use free PLOP software (below) to design your mirror cell.
  • PLOP Automated Mirror Cell Optimization. This free windows software can help you design a “perfect” mirror cell. Just plug in the numbers, and in seconds, you have a mirror cell design.
  • Engineering, Design and Construction of Portable Newtonian Telescopes, Albert Highe. Do you want your next telescope to truly satisfy you? This book dedicates an entire chapter that asks you questions that help you design and build (or buy!) a telescope that will do just that. And it has beautifully engineered contemporary designs for large truss telescopes.
  • Engineering, Design and Construction of String Telescopes, Albert Highe. Beautifully engineered, yet challenging, ultra-light, air transportable newtonian telescope designs.
  • Newt for the Web (Stellafane). This is a simple, yet effective tool for newtonian telescope design. You can design an excellent telescope with just this free tool, plus old school drafting tools like ruler, protractor, pencil and compass.
  • Reflecting Telescope Optimizer Suite. Mel Bartels created this wonderful free online newtonian telescope design tool: If you explore Mel’s website you will find innovative, ultra-fast dobsonian telescopes, beautiful deep sky sketches, and mind expanding ideas that will probably make you a better observer.
  • Right Angled Triangles Calculator, Cleve Books. Are you building a truss telescope but can’t remember trigonometry? This site makes it easy:
  • Stargazer Steve 6” Truss Telescope. A very portable, ultra-light commercial truss telescope. Moderately priced, too!
  • Explore Scientific 8”f3.9 Want a fast scope but don’t want to build it? This fast astrograph optical tube assembly has a carbon fibre tube and weighs 18.3 pounds / 8.3kg. It’s remarkably affordable, too!
  • Orion 8” f/3.9 You can save a lot of work by buying a telescope off the shelf, like this one. Similar to the Explore Scientific, but with a steel tube at an irresistable price. And this OTA is under 18 pounds / 8kg!
  • R. F. Royce Telescope Building Projects. Simple newtonian telescope designs by one of the finest opticians on planet Earth. The first telescope I built, a 10”f6, and the second telescope I built, a 6”f8, were both based on Royce’s designs. Both performed far beyond my expectations. In fact, the surrier-trusses for my 8”f4 design were based on the Royce design. Want to build your ultimate lunar and planetary telescope? Click the third link. And… considering how much you can learn from one of the world’s greatest opticians, shouldn’t you click every link?
  • Reiner Vogel Travel Dobs. If you are interested in designing and building your own telescope, have a look at this website. You will find easy and effective construction techniques and ultralight, ultra-portable telescopes here. And big ones. You’ll find equatorial mounts and observing notes, too!
  • Here is my talk at the RASC, Toronto, (Royal Astronomical Society of Canada) entitled, “Designing and Building a Newtonian Telescope for Wide Field Visual and Air Travel”. You can scroll the video to 38:20 if you want to go directly to my presentation.

Easy ways to show the earth is not flat


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Sharing Steve Ruis’ handy methods anybody can use to show that the earth is in fact round. Very clever!

An Open Letter to the Many Flat Earthers Now in Existence

by Steve Ruis

Dear Flat Earthers,
Many people have been derogatory of your belief that the Earth is flat. Please note that they are belittling your belief, not you per se. You, personally, are an idiot, but that is probably not your fault.

Here are any number of accessible approaches for discovering the shape of our beloved planet. Enjoy!

* * *

Use Your Phone!
On Christmas Day, here in Chicago, I expect there to be snow on the ground because, well, it is winter. On Christmas Day I can pick up my phone and dial up anyone in Australia and ask them “What season is it?” They will tell you that it is summer in Australia. You might want to ask your flat Earth mentors how it could be winter and summer simultaneously on a flat Earth.

Use Your Phone!
Go to a globe and pick a spot half way around the Earth (I know it is a false representation in your belief, but humor me.) In the middle of the day, phone somebody at or near that spot. Call a hotel, they are always open. Ask whoever responds “Is it light or dark outside?” They will tell you that it is dark where they are. You might want to ask your flat Earth mentors how it could be light and dark simultaneously on a flat Earth.

Look Up What Local Time Was
In the US there was this concept of “local time” which was that “noon” was when the sun was at its highest point in its arc. You could call up people on the telephone who were not that far away and ask them what time it was and they would tell you something different from what your clock was telling you. The farther away they were, the greater the difference would be. On a flat Earth the time would be the same everywhere.

Look Up What Time Zones Are
I am writing this in the central time zone in the U.S. These zones were created at the behest of the railroad industry whose dispatchers were going crazy making up schedules for trains when every place had their own times. By creating these “zones” everything would be exactly one hour off from those in neighboring zones, two hours off for the next over zones, and so on. If you don’t believe me . .  pick up your phone and dial up a friend who lives a considerable distance (east-west) away from you and ask them what time it is. The time they state will be a whole number of hours away from your time. Heck, even the NFL knows this. When I lived on the left coast, the games started at 10 AM and 1 PM. Now that I live in the central time zone, the games start at 12 Noon and 3 PM. Over New York way the games start at 1PM and 4 PM. Do you think those games are replayed in one hour increments? Nope, time zones!. You might want to ask your flat earth mentors how it could be that simultaneous games start at different times on a flat Earth.

Watch the Video
Astronauts in the International Space Station (ISS) have made continuous videos of an entire orbit of the Earth. It takes only about an hour and a half about the length of a typical Hollywood movie. During the whole movie the earth appears round, and yet it is clear that different continents are passing in our view.

Now you may argue that NASA made this movie as propaganda for the Round Earth Conspiracy. It is certainly within our CGI abilities at this point, but you may want to ask why NASA would want to do such a thing? Plus, many astronauts have taken their own cameras aboard and taken pictures for themselves and they show the same thing. How could the Round Earth Conspiracy have allowed that to happen? It must be incompetence! Conspiracies aren’t what they used to be!

Da Balloon, Boss, Da Balloon
Many amateurs, unaffiliated with the government, have launched rockets and balloons high up into the atmosphere to take pictures. Every damned one of those pictures shows that the Earth is round. How come all of those cameras ended up pointed at the curved edge of your round and flat disk Earth? Such a coincidence!

An Oldie But Goodie #1
Occasionally, during a lunar eclipse, you can see the shadow of the earth falling upon the Moon. The shadow is always circular. This would be true if the flat earth were always dead on to the Moon, but the Moon orbits the Earth and wouldn’t a flat Earth be edgewise, often as not, and wouldn’t that create a non-round shadow on the Moon? Inquiring minds want to know.

An Oldie But Goodie #2
It was claimed that one of the first demonstrations of the earth being round was the observation of ships sailing west from Europe/England could be observed for a while but the ship itself was lost to sight while the mast was still visible. This would not happen on a flat Earth. The whole ship would just get smaller and smaller as it sailed west.

For pity’s sake, I live 22 stories up and the shores of Lake Michigan and I cannot see anything directly opposite me in Michigan. All I can see is water, with any kind of magnification I can muster. And I am not looking across the widest part of this lake! If the earth were flat, the lake would be flat and I could see the Michigan shore.

And Finally . . .

All of the fricking satellites! Do the math. What kind of orbit is stable around a flat disk earth? Answer none! And there are hundreds of the danged things in orbit.

Also, just for giggles. Look up what a Foucault pendulum is, And explain its behavior based upon a flat Earth.

PS You may be getting good vibes in your special knowledge that you know something other people do not. However, would not that special feeling be more worthwhile were you to volunteer at a food bank or a day care center or senior center? Wouldn’t doing something worthwhile be more rewarding that making a statement about how those pointy-headed intellectuals aren’t so smart?

PPS I have seen the cute models with the Sun and Moon on sticks rotating around (see photo above). If that were the case, everyone could see the Sun and Moon all day, every day. (There is straight line access to both objects in that model from everywhere on the flat disk.) Do you see the Sun and Moon all day, every day? No? Maybe someone who had more creativity than knowledge came up with those models. They do sell well, I must admit, so maybe their interest is commercial.

PPPS Regarding the 200 foot wall of ice that supposedly exists at the “edge of the disk,” supposedly so all the water doesn’t flow off and be lost into space. By now don’t you think someone would have sailed next to that wall all of the way? That distance would be somewhere in the neighborhood of a 28,000 mile trip. Has anyone ever report such a thing? Hmm, I wonder why not

SOLD: Antique 6″ f/14 Refractor With Good Optics Available No Longer


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The Hopewell Observatory had available a finely-machined antique, brass-tube 6″ f./14 achromatic refractor.

The mount and drive were apparently made by John Brashear, but we don’t know for sure who made the tube, lens, focuser or optics.

We removed a lot of accumulated green or black grunge on the outside of the tube, but found no identifying markings of any sort anywhere, except for the degrees and such on the setting circles and some very subtle marks on the sides of the lens elements indicating the proper alignment.

The son of the original owner told me that the scope and mount were built a bit over a century ago for the American professional astronomer Carl Kiess. The latter worked mostly on stellar and solar spectra for the National Bureau of Standards, was for many years on the faculty of Georgetown University, and passed away in 1967. A few decades later, his son later donated this scope and mount to National Capital Astronomers (of DC), who were unable to use it. NCA then later sold it to us (Hopewell Observatory), who cleaned and tested it.

The attribution of the mount to Brashear was by Bart Fried of the Antique Telescope Society, who said that quite often Brashear didn’t initial or stamp his products. Looking at known examples of Brashear’s mounts, I think Fried is probably correct. Kiess’s son said he thought that the optics were made by an optician in California, but he didn’t remember any other details. His father got his PhD at UC Berkeley in 1913, and later worked at the Lick Observatory before settling in the DC area. The company that Brashear became doesn’t have any records going back that far.

When we first looked through the scope, we thought the views were terrible, which surprised us. However, as we were cleaning the lens cell, someone noticed subtle pencil marks on the edges of the two lens elements, indicating how they were supposed to be aligned with each other. Once we fixed that, and replaced the 8 or so paper tabs with three blue tape tabs, we found it produced very nice views indeed!

The focuser accepts standard 1.25″ eyepieces, and the focuser slides very smoothly (once we got the nasty, flaky corrosion off as delicately as possible and sprayed the metal with several coats of clear polyurethane). The workmanship is beautiful!

Top: tiller for hand control of right ascension. Middle: counterweight bar (machined by me to screw into the mount) with clamps to hold weights in place. Bottom: detail of 1.25″ rack-and-pinion focuser.

We have not cleaned the mechanical mount, or tried it out, but it does appear to operate: the user turns a miniature boat tiller at the end of a long lever to keep up with the motions of the stars.

The mount and cradle (with size 12 feet for scale)

The counterweight rod was missing, so I machined a replacement, which has weight holder clamps like you see in gymnasiums. Normal Barbell-type weights with 1 inch holes fit well and can be adjusted with the clamps.

Unfortunately, the whole device is rather heavy, and we already own a nice 6″ f/15 refractor made by Jaegers, as well as some Schmidt-Cassegrain telescopes that also have long focal lengths. Putting this scope on its own pedestal, outside our roll-off roof, with adequate protection from both the elements and from vandals, or figuring out a way to mount it and remove it when needed, are efforts that we don’t see as being wise for us.

Did I mention that it’s heavy? The OTA and the mount together weigh roughly 100 pounds.

However, it’s really a beautiful, historic piece with great optics. Perhaps a collector might be interested in putting this in a dome atop their home or in their office? Or perhaps someone might be interested in trading this towards a nice Ritchey Chretien or Corrected Dal-Kirkham telescope of moderate aperture?

Anybody know what might be a fair price for this?

Guy Brandenburg


The Hopewell Observatory

Some more photos of the process and to three previous posts on this telescope.

Partway through cleaning the greenish, peeling, grimy layer and old duct tape residue with a fine wire brush at low speed to reveal the beautiful brass OTA.
This shows the universal joint that attaches to the ’tiller’ and drives the RA axis
Do you see the secret mark, not aligned with anything?
Aluminum lens cover and cell before cleaning
Lens cell and cover, with adjustment screws highlighted, after cleaning
It works!

Some WW2 or Cold-War-Era Aerial Surveillance Cameras


(Think U2 spyplanes.. )

Hopewell Observatory has three WW2 or Cold-War aerial spy camera optical tube assemblies, including a relatively famous Fairchild K-38. No film holders, though. And no spy planes. The lenses are in good condition, and the shutters seem to work fine.

We would like to give them away to someone who wants and appreciates them, and can put them to good use. Does anybody know someone who would be interested?

They’ve been sitting unused in our clubhouse for over 20 years. Take one, take two, take all of them, we want them gone.

We are located in the DC / Northern Virginia area. Nearby pickup is best. Anybody who wants them shipped elsewhere would obviously need to pay for packaging and shipping.

Here are some photos.

This one is labeled K-38, has a special, delicate, fluorite lens in front, and is stamped with the label 10-10-57 – perhaps a date. The shoe is for scale.


The next two have tape measures and shoes for scale.


Let me know (a comment will work) if you are interested.

Can Mathematicians be Replaced by Computers?


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The short answer is, certainly not yet.

Can they ever be? From reading this article and my own experience with the geometry-proving-and-construction software called Geometrix, written by my friend and colleague Jacques Gressier, I am not sure it’s possible at all.

Here is an interesting article that I’m copying and pasting from Jerry Becker at SIU, who got it from Quanta:

From Quanta, Thursday, August 27, 2020. SEE
How Close Are Computers to Automating Mathematical Reasoning?
AI tools are shaping next-generation theorem provers, and with them the relationship between math and machine.
By Stephen Ornes
In the 1970s, the late mathematician Paul Cohen, the only person to ever win a Fields Medal for work in mathematical logic, reportedly made a sweeping prediction that continues to excite and irritate mathematicians – that “at some unspecified future time, mathematicians would be replaced by computers.” Cohen, legendary for his daring methods in set theory, predicted that all of mathematics could be automated, including the writing of proofs.

A proof is a step-by-step logical argument that verifies the truth of a conjecture, or a mathematical proposition. (Once it’s proved, a conjecture becomes a theorem.) It both establishes the validity of a statement and explains why it’s true. A proof is strange, though. It’s abstract and untethered to material experience. “They’re this crazy contact between an imaginary, nonphysical world and biologically evolved creatures,” said the cognitive scientist Simon DeDeo of Carnegie Mellon University, who studies mathematical certainty by analyzing the structure of proofs. “We did not evolve to do this.”

Computers are useful for big calculations, but proofs require something different. Conjectures arise from inductive reasoning – a kind of intuition about an interesting problem – and proofs generally follow deductive, step-by-step logic. They often require complicated creative thinking as well as the more laborious work of filling in the gaps, and machines can’t achieve this combination.

Computerized theorem provers can be broken down into two categories. Automated theorem provers, or ATPs, typically use brute-force methods to crunch through big calculations. Interactive theorem provers, or ITPs, act as proof assistants that can verify the accuracy of an argument and check existing proofs for errors. But these two strategies, even when combined (as is the case with newer theorem provers), don’t add up to automated reasoning.

SIDEBAR PHOTO:  Simon DeDeo of Carnegie Mellon helped show that people and machines seem to construct mathematical proofs in similar ways. Courtesy of Simon DeDeo
Plus, the tools haven’t been met with open arms, and the majority of mathematicians don’t use or welcome them. “They’re very controversial for mathematicians,” DeDeo said. “Most of them don’t like the idea.”

A formidable open challenge in the field asks how much proof-making can actually be automated: Can a system generate an interesting conjecture and prove it in a way that people understand? A slew of recent advances from labs around the world suggests ways that artificial intelligence tools may answer that question. Josef Urban at the Czech Institute of Informatics, Robotics and Cybernetics in Prague is exploring a variety of approaches that use machine learning to boost the efficiency and performance of existing provers. In July, his group reported a set of original conjectures and proofs generated and verified by machines. And in June, a group at Google Research led by Christian Szegedy posted recent results from efforts to harness the strengths of natural language processing to make computer proofs more human-seeming in structure and explanation.

Some mathematicians see theorem provers as a potentially game-changing tool for training undergraduates in proof writing. Others say that getting computers to write proofs is unnecessary for advancing mathematics and probably impossible. But a system that can predict a useful conjecture and prove a new theorem will achieve something new –  some machine version of understanding, Szegedy said. And that suggests the possibility of automating reason itself.
Useful Machines

Mathematicians, logicians and philosophers have long argued over what part of creating proofs is fundamentally human, and debates about mechanized mathematics continue today, especially in the deep valleys connecting computer science and pure mathematics.

For computer scientists, theorem provers are not controversial. They offer a rigorous way to verify that a program works, and arguments about intuition and creativity are less important than finding an efficient way to solve a problem. At the Massachusetts Institute of Technology, for example, the computer scientist Adam Chlipala has designed theorem-proving tools that generate cryptographic algorithms – traditionally written by humans – to safeguard internet transactions. Already, his group’s code is used for the majority of the communication on Google’s Chrome browser.

SIDEBAR PHOTO: Emily Riehl of Johns Hopkins University uses theorem provers in teaching students and proof assistants in her own work. “Using a proof assistant has changed the way I think about writing proofs,” she said. Will Kirk/Johns Hopkins University
“You can take any kind of mathematical argument and code it with one tool, and connect your arguments together to create proofs of security,” Chlipala said.

In math, theorem provers have helped produce complicated, calculation-heavy proofs that otherwise would have occupied hundreds of years of mathematicians’ lives. The Kepler conjecture, which describes the best way to stack spheres (or, historically, oranges or cannonballs), offers a telling example. In 1998, Thomas Hales, together with his student Sam Ferguson, completed a proof using a variety of computerized math techniques. The result was so cumbersome – the results took up 3 gigabytes – that 12 mathematicians analyzed it for years before announcing they were 99% certain it was correct.

The Kepler conjecture isn’t the only famous question to be solved by machines. The four-color theorem, which says you only need four hues to color any two-dimensional map so that no two adjoining regions share a color, was settled in 1977 by mathematicians using a computer program that churned through five-colored maps to show they could all be reduced to four. And in 2016, a trio of mathematicians used a computer program to prove a longstanding open challenge called the Boolean Pythagorean triples problem, but the initial version of the proof was 200 terabytes in size. With a high-speed internet connection, a person could download it in a little over three weeks.

Complicated Feelings
These examples are often trumpeted as successes, but they’ve also added to the debate. The computer code proving the four-color theorem, which was settled more than 40 years ago, was impossible for humans to check on their own. “Mathematicians have been arguing ever since whether or not it’s a proof,” said the mathematician Michael Harris of Columbia University.
SIDEBAR PHOTO:  Many mathematicians, like Columbia University’s Michael Harris, disagree with the idea that computerized theorem provers are necessary – or that they’ll make human mathematicians obsolete. Béatrice Antolin
Another gripe is that if they want to use theorem provers, mathematicians must first learn to code and then figure out how to express their problem in computer-friendly language – activities that detract from the act of doing math. “By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself,” Harris said.

Many just don’t see a need for theorem solvers in their work. “They have a system, and it’s pencil and paper, and it works,” said Kevin Buzzard, a mathematician at Imperial College London who three years ago pivoted his work from pure math to focus on theorem provers and formal proofs. “Computers have done amazing calculations for us, but they have never solved a hard problem on their own,” he said. “Until they do, mathematicians aren’t going to be buying into this stuff.”

But Buzzard and others think maybe they should. For one thing, “computer proofs may not be as alien as we think,” DeDeo said. Recently, together with Scott Viteri, a computer scientist now at Stanford University, he reverse-engineered a handful of famous canonical proofs (including one from Euclid’s Elements) and dozens of machine-generated proofs, written using a theorem prover called Coq, to look for commonalities. They found that the networked structure of machine proofs was remarkably similar to the structure of proofs made by people. That shared trait, he said, may help researchers find a way to get proof assistants to, in some sense, explain themselves.
“Machine proofs may not be as mysterious as they appear,” DeDeo said.

Others say theorem provers can be useful teaching tools, in both computer science and mathematics. At Johns Hopkins University, the mathematician Emily Riehl has developed courses in which students write proofs using a theorem prover. “It forces you to be very organized and think clearly,” she said. “Students who write proofs for the first time can have trouble knowing what they need and understanding the logical structure.”

SIDEBAR:  By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself.  Michael Harris, Columbia University
Riehl also says that she’s been increasingly using theorem provers in her own work. “It’s not necessarily something you have to use all the time, and will never substitute for scribbling on a piece of paper,” she said, “but using a proof assistant has changed the way I think about writing proofs.”

Theorem provers also offer a way to keep the field honest. In 1999, the Russian American mathematician Vladimir Voevodsky discovered an error in one of his proofs. From then until his death in 2017, he was a vocal proponent of using computers to check proofs. Hales said that he and Ferguson found hundreds of errors in their original proof when they checked it with computers. Even the very first proposition in Euclid’s Elements isn’t perfect. If a machine can help mathematicians avoid such mistakes, why not take advantage of it? (The practical objection, justified or not, is the one suggested by Harris: If mathematicians have to spend their time formalizing math to be understood by a computer, that’s time they’re not spending doing new math.)

But Timothy Gowers, a mathematician and Fields medalist at the University of Cambridge, wants to go even further: He envisions a future in which theorem provers replace human referees at major journals. “I can see it becoming standard practice that if you want your paper to be accepted, you have to get it past an automatic checker,” he said.

But before computers can universally check or even devise proofs, researchers first have to clear a significant hurdle: the communication barrier between the language of humans and the language of computers.

Today’s theorem provers weren’t designed to be mathematician-friendly. ATPs, the first type, are generally used to check if a statement is correct, often by testing possible cases. Ask an ATP to verify that a person can drive from Miami to Seattle, for example, and it might search all cities connected by roads leading away from Miami and eventually finding a city with a road leading into Seattle.

SIDEBAR PHOTO: Not every mathematician hates theorem provers. Timothy Gowers, of the University of Cambridge, thinks they may one day replace human reviewers at mathematical journals. The Abel Prize
With an ATP, a programmer can code in all the rules, or axioms, and then ask if a particular conjecture follows those rules. The computer then does all the work. “You just type in the conjecture you want to prove, and you hope you get an answer,” said Daniel Huang, a computer scientist who recently left the University of California, Berkeley, to work at a startup.

But here’s the rub: What an ATP doesn’t do is explain its work. All that calculating happens within the machine, and to human eyes it would look like a long string of 0s and 1s. Huang said it’s impossible to scan the proof and follow the reasoning, because it looks like a pile of random data. “No human will ever look at that proof and be able to say, ‘I get it,'” he said.

ITPs, the second category, have vast data sets containing up to tens of thousands of theorems and proofs, which they can scan to verify that a proof is accurate. Unlike ATPs, which operate in a kind of black box and just spit out an answer, ITPs require human interaction and even guidance along the way, so they’re not as inaccessible. “A human could sit down and understand what the proof-level techniques are,” said Huang. (These are the kinds of machine proofs DeDeo and Viteri studied.)
ITPs have become increasingly popular in recent years. In 2017, the trio behind the Boolean Pythagorean triples problem used Coq, an ITP, to create and verify a formal version of their proof; in 2005 Georges Gonthier at Microsoft Research Cambridge used Coq to formalize the four-color theorem. Hales also used ITPs called HOL Light and Isabelle on the formal proof of the Kepler conjecture. (“HOL” stands for “higher-order logic.”)

Efforts at the forefront of the field today aim to blend learning with reasoning. They often combine ATPs with ITPs and also integrate machine learning tools to improve the efficiency of both. They envision ATP/ITP programs that can use deductive reasoning – and even communicate mathematical ideas – the same way people do, or at least in similar ways.

The Limits of Reason

Josef Urban thinks that the marriage of deductive and inductive reasoning required for proofs can be achieved through this kind of combined approach. His group has built theorem provers guided by machine learning tools, which allow computers to learn on their own through experience. Over the last few years, they’ve explored the use of neural networks – layers of computations that help machines process information through a rough approximation of our brain’s neuronal activity. In July, his group reported on new conjectures generated by a neural network trained on theorem-proving data.

Urban was partially inspired by Andrej Karpathy, who a few years ago trained a neural network to generate mathematical-looking nonsense that looked legitimate to nonexperts. Urban didn’t want nonsense, though – he and his group instead designed their own tool to find new proofs after training on millions of theorems. Then they used the network to generate new conjectures and checked the validity of those conjectures using an ATP called E.

The network proposed more than 50,000 new formulas, though tens of thousands were duplicates. “It seems that we are not yet capable of proving the more interesting conjectures,” Urban said.

SIDEBAR: [Machines] will learn how to do their own prompts.  Timothy Gowers, University of Cambridge
Szegedy at Google Research sees the challenge of automating reasoning in computer proofs as a subset of a much bigger field: natural language processing, which involves pattern recognition in the usage of words and sentences. (Pattern recognition is also the driving idea behind computer vision, the object of Szegedy’s previous project at Google.) Like other groups, his team wants theorem provers that can find and explain useful proofs.

Inspired by the rapid development of AI tools like AlphaZero – the DeepMind program that can defeat humans at chess, Go and shogi – Szegedy’s group wants to capitalize on recent advances in language recognition to write proofs. Language models, he said, can demonstrate surprisingly solid mathematical reasoning.

His group at Google Research recently described a way to use language models – which often use neural networks – to generate new proofs. After training the model to recognize a kind of treelike structure in theorems that are known to be true, they ran a kind of free-form experiment, simply asking the network to generate and prove a theorem without any further guidance. Of the thousands of generated conjectures, about 13% were both provable and new (meaning they didn’t duplicate other theorems in the database). The experiment, he said, suggests that the neural net could teach itself a kind of understanding of what a proof looks like.

“Neural networks are able to develop an artificial style of intuition,” Szegedy said.

Of course, it’s still unclear whether these efforts will fulfill Cohen’s prophecy from over 40 years ago. Gowers has said that he thinks computers will be able to out-reason mathematicians by 2099. At first, he predicts, mathematicians will enjoy a kind of golden age, “when mathematicians do all the fun parts and computers do all the boring parts. But I think it will last a very short time.”
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After all, if the machines continue to improve, and they have access to vast amounts of data, they should become very good at doing the fun parts, too. “They will learn how to do their own prompts,” Gowers said.

Harris disagrees. He doesn’t think computer provers are necessary, or that they will inevitably “make human mathematicians obsolete.” If computer scientists are ever able to program a kind of synthetic intuition, he says, it still won’t rival that of humans. “Even if computers understand, they don’t understand in a human way.”