Michael A. Covington    Michael A. Covington, Ph.D.
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Popular topics on this page:
Periodic-error correction
What is 6÷2(1+2)? (Not a math test)
Why people can't spell lens
50 years of amateur astronomy
Will sexual harassment finally go out of style?
Making the alphabet legible, German-style
What I saw 50 years ago
The finest sights I've seen in the sky
R, non-Latin alphabets, Linux, and VirtualBox
M31 (Andromeda Galaxy)
Central Cygnus
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95 + 500

On the 500th anniversary of Luther's 95 Theses, I invite my fellow Christians, Protestant, Catholic, and Orthodox, to join me in praying for Christian unity and faithfulness to divine revelation.


R, non-Latin alphabets, and why (and how) I must use Linux

Very seldom do I actually have to use an operating system other than Windows, except of course when developing softare for someone else to run under another operating system. But one of those occasions has come. R doesn't handle non-Latin alphabets properly under Windows, as far as I can tell.

First, the good news: The best way to run Linux with Windows, in my experience, is to install VirtualBox (free) and then install 64-bit Linux Mint MATE to run under it. If VirtualBox doesn't want to host 64-bit operating systems, check that Intel virtualization is turned on in your BIOS and Hyper-V is turned off under Windows features. Then designate your main Windows directory (\users\name) as a "shared folder" that the VirtualBox Linux VM can see, permanent, auto-mount, and reboot Linux.

I can get Linux up and running in 40 seconds, and one of my Windows 10 windows is then full-screen Linux. I can Alt-Tab between it and the various Windows windows.

Previous alternatives that I tried were Microsoft's Bash for Windows (mixes very well with Windows but has no GUI, and had problems with Java the last time I tried it) and simply dual-booting Windows and Linux (which gave me no access to either OS while running the other one).

Now then. What was the problem? In the hope that some R guru out there can solve it, let me offer a demonstration. I had to concoct one in order not to show any of my client's data or even what language is involved.

Click here for a test data file. It is a CSV file, encoded in UTF-8 with BOM, and opens just fine in Excel:

What I challenge you to do is open it in your favorite interactive R environment and execute these commands:

That screenshot is from RStudio under Linux. As you can see, the Greek came through just fine. It also looks fine in View and, most importantly, in graphics:

I had no such luck under Windows. You can see on the console display that Windows has read in the Unicode characters but cannot write them back out correctly, nor display them in graphics. (It can write them back out to a text file; that much I'll grant.) This is RStudio, but I had the same problems under the R GUI and in the R environment in Visual Studio.

The alert reader will see that FEFF was treated as an ordinary character. I tried creating a data file in UTF-8 without BOM, and the only difference it made was that the FEFF wasn't there.

I have looked in vain for a setting in RStudio that would fix this. I would appreciate knowing about it if there is one. Please note that I am looking for a reliable solution for production work, not a kluge or tour de force.


What I saw 50 years ago

Continuing to commemorate my 50 years of amateur astronomy, the other night I took a picture to show you what I viewed through binoculars that night half a century ago. You're looking at central Cygnus. I particularly remember the star cloud at the bottom just left of center.

This is a stack of 17 1-minute exposures with a Sigma 90-mm f/2.8 lens at f/4 and a Nikon D5300 on my iOptron SkyTracker. The lens is one I haven't used much for astrophotography; I bought it to use with my Nikon N70 back in the film era. With the D5300, it doesn't autofocus but does allow the exposure meter to be used (no help with a picture like this, of course).

The picture was taken in town, and I was observing in the country (50 years ago). But the camera sees more than the eye, so it balances out.

The finest sights I have seen in the sky

The following are some of the most memorable things I've seen in the sky during my fifty years of observing. Surprisingly many of them were visible to the naked eye.

Deep red lunar eclipse, April, 1968.

Total solar eclipse (under clouds), March 7, 1970.

Weather on Saturn, 1970-71. I see no published record of a white spot that year, but I definitely observed something of the sort, at least some turbulence, with my Criterion RV-6 telescope, and made many drawings.

Milky Way with Centaurus and Crux, high overhead, from Australia, 1973. Not a dark country sky, but an impressive view nonetheless.

Aurora borealis from an airliner over Newfoundland, 1980.

Comet Halley, 1986. I regret that I did not get a very good picture of it.

Spots on Jupiter caused by Comet Shoemaker-Levy, 1994. The comet hit the planet.

Comets Hale-Bopp and Hyakutake, 1996-1997. Well observed and well photographed.

Leonid meteor storms, 2001 and 2002. A once-in-a-century experience, twice!

Aurora borealis (from Georgia!), 2003. A display that would have been decent even by Scandinavian standards, but I saw it from the Deep South.

Many fine views of the summer Milky Way from Deerlick, 2007-present.

Total solar eclipse, August 21, 2017. Under a clear sky.

Of course, what I enjoy most is viewing the richness of the starry sky and the variety of objects in it, including stars, clusters, nebulae, and galaxies. But those objects are constantly present, and I have seen many of the finest ones hundreds of times.

Losing my amateur status?

I'm getting ready to attend the Georgia Regional Astronomers' Meeting, a (mostly) professional meeting, and will be giving a poster presentation about DSLR sensor characteristics.

That, and Digital SLR Astrophotography is (finally) coming along at a brisk pace... so I'll be doing things other than writing blog entries for a while. I'll be back at least once before November, though.


Making the alphabet more legible

With the Fall Creators Update, the huge Windows 10 update that is being delivered now, Microsoft has given us a typeface called Bahnschrift with some distinct advantages. Here's what it looks like:

This is actually a derivative of the DIN typefaces that the Germans have been refining for decades for use on road signs and equipment labels.

And maybe it will free us from the grip of the Helvetica family of fonts.

Let me explain. I always thought the point of an alphabet was to make all the letters look different from each other. But Helvetica and its close cousin Arial, so pervasive during my lifetime, don't do that. They make the lowercase L and the uppercase I look just alike, as you can see in the word "Illinois."

Gill Sans, popular in Britain, even makes the digit 1 (one) look like I and L too.

Everyone agrees that serif typefaces like Times New Roman are more legible under normal conditions. That's why we use them for books and newspapers. But on road signs and equipment labels, we prefer sans-serif type because it holds up better when printed at low resolution or not seen clearly. (Times New Roman loses a lot when even slightly blurred.)

Some time ago I wrote about how, when it was new, Helvetica had an air of modernity, but nowadays it's just ordinary type. Ordinary type with a legibility problem because the I and the lowercase L look alike except for a very slight difference of weight and spacing. Newer fonts (Myriad, Segoe) improve the balance of the curves but keep the basic letterforms of Helvetica, including the I-L conflation.

That's the problem Bahnschrift solves, even though its original design is actually older than Helvetica.

In a world in which more and more of our writing is not just plain English, but includes things such as web addresses and other computer codes, we need to be able to tell what every single letter is, without relying on context.

I won't be using Bahnscrift to label the illustrations in my next book; it would clash too much with Helvetica derivatives already in use there. But I'm going to start using it for labels on equipment and maybe even for a conference poster. I'm the same age as Helvetica (1957); maybe I'll outlive it.


Off to Andromeda!

This is my first astronomical photograph in several weeks. I'm spending October having bronchitis and, apparently, missing the Peach State Star Gaze, but I did get a nice picture of the galaxies M31, M32 (below center), and M110 (top) last night (October 17).

This was going to be just an equipment-testing session, but the pictures came out well. This is a stack of 6 3-minute exposures through an AT65EDQ refractor (65-mm f/6.5) and Nikon D5300 camera body on an AVX mount, autoguided with a 60-mm guidescope, DMK camera, and PHD2 guiding software.

The air was unusually clear, and I could almost see M31 with the unaided eye at my home in Athens, Georgia.

A good source for computer accessories

In taking the picture above, the autoguiding was controlled by a small Asus laptop. To operate the laptop from my portable battery system, I needed a 12-volt car power supply for it, so a few months ago, I bought one. I want to recommend an Amazon vendor who is a particularly good supplier of such things and has given me excellent service: EVASALES101 via Amazon.

That's the place to go for accessories for computers that are a couple of years old, whose accessories are no longer in the big-box stores.


Will sexual harassment finally go out of style?

[Minor revisions.]

I am heartened by the way our popular culture is, at least temporarily, coming to its senses about sexual harassment and exploitation of women.

First Hugh Hefner died, and — surprise! — he wasn't hailed as a hero of women's liberation, but rather as a villain of their exploitation. Almost nothing good has been said about him.

Then Harvey Weinstein's misdeeds caught up with him, and apparently he's the first of several high-profile cases. And the "Me, too" campaign has encouraged women to tell their stories. Many of us are aghast at the amount of harassment that almost all women have had to put up with in civilized environments. It ranges from catcalls and inappropriate flirting all the way up to — surprisingly often — threats of rape.

Maybe people are coming to their senses. Maybe this is more than a flash in the pan. Let's make some lists.

Things that help solve the problem:

(1) High-profile cases of men losing their jobs and careers because they have harassed and abused women.

This is the only example that a lot of people will learn from. Not being an abusive man, I don't entirely know how their minds work, but I think what has happened is that they live in a fantasy world — they have tuned out almost all warnings that what they are doing is wrong.

But when they see careers come crashing down, and see that even the rich and famous are not immune — that they are condemned as exploiters, not admired as playboys — maybe they'll listen.

(2) Women, men, parents, and friends who make it crystal-clear that harassment is wrong.

People just aren't getting the message. We need to say it louder.

And some men are genuinely mistaken about what (most) women consider flattering vs. annoying. No, they don't enjoy catcalls or double-entendre flirtation from strangers or near-strangers.

(3) Women who are willing to object loudly and strongly when they can — verbally, administratively, legally, or even physically.

My daughters have been taught to fight to maim if physically abused. Or else bring legal penalties. Or both.

(4) Women high up in the management chain in businesses and organizations.

Harassment happens particularly when all the women are subordinate to all the men. Let's get away from that.

(5) Men in management positions who make it clear that sexual harassment is intolerable and the guilty will take the consequences of their actions.

That's the main thing I've been able to do, in years of having some administrative responsibility at the University of Georgia. I never handled a serious case of sexual harassment, but I probably prevented some, working with my colleagues. We made sure the right warnings were given at each year's orientation and that people knew where we stood.

(6) Legal departments and liability insurance carriers that care.

That's how we got rid of secondhand smoke, after all; maybe it's how we can get rid of a worse pest.

Things that make the problem worse:

(1) Abusive men, and other men who tolerate and encourage them.

If your friend threatens to commit a crime (yes, sexual assault is a crime), don't chuckle or grin. Warn him the same way you'd warn him if he were about to rob a bank. And if he is about to make a pest of himself, clue him in that it's not a wise thing to do.

(2) Those who ignore the warning signs.

Harassment is usually a long-established habit. If you see or hear evidence that something improper is going on, and you are in a situation to discourage it or to bring it to the right person's attention, you have a duty to do so.

(3) Women who "play the game."

I've encountered them — women who want to go over the line because they think there will be rewards. (Think of the Hollywood casting couch.) They think they can manipulate the men who are trying to manipulate them.

These are not numerous, but some of them speak loudly, and say, in essence, "You have to play the game to get ahead. I played the game and benefited from it. If you're not willing to play the game, get out of the workplace."

Bah humbug!

(4) Cynics. "It will always be that way, there's nothing you can do..."

Cynics are people who brag about how bad things are — who feel superior because they have seen through our naive idealism, they see that our attempts to make things better are futile, they "see through" everything.

C. S. Lewis pointed out that to "see through" all things is the same as not to see them.

(5) Entertainment that spreads misunderstanding.

I have long distrusted "men's entertainment" and the men who will believe a fantasy story in a magazine rather than the plain words that women speak to them.

(6) Anyone who discourages others from making the situation better.

As a chemistry professor once said, "If you're not part of the solution, you're part of the precipitate."

Red herrings (things that sidetrack us):

(1) Prudery.

The notion that sexuality is shameful leads people to see wrongdoing where there is none. And then they're out of touch with reality. "He was alone with her for 15 minutes; they must be up to some hanky-panky." No; they were cleaning out the stockroom.

Being out of touch with reality is not a virtue.

(2) Pouncing on momentary fumbles.

Anyone can momentarily say or do something that is open to misinterpretation. We don't want to crucify people for accidental speech errors. (As a linguist, I know better than most people that talking is a demanding activity, and people blunder while doing it.)

Abuse comprises either words or acts that are clearly over the line, or consistent patterns of behavior. And those patterns are often very long-standing before someone is held responsible.

(3) Gossip.

Spreading gossip about people's alleged abusive behavior can be very destructive. If you have concrete grounds for suspicion, bring them to the right person, someone with administrative responsibility, if possible. Don't spread it to your friends at random. Victims don't want their stories publicized, often inaccurately, by third parties.

There is such a thing as false accusation. There are people who want to ruin others' reputations with lies. Don't let them use you. That it itself a form of abuse.

I should add that allegations should be investigated fairly. There should never be an expectation that people will believe allegations on scant testimony without investigation. But real cases of abuse are common and are often easy to substantiate, especially since one perpetrator usually has many victims.


An exceptionally good penlight

A couple of months ago I bought myself a highly recommended penlight, and now I find myself using it constantly. It's amazing how many holes, nooks, and crannies in the modern world need a bit of extra light in them.

The ThruNite Ti4T is no ordinary penlight. It uses a white LED and has three brightness levels, the lowest one penlight-like, the other two very bright and even brighter. On top of that, it has a flashing mode that can be used for emergency signaling. All these are chosen simply and easily by pressing the button repeatedly.

This is, in effect, a penlight that can be as bright as a full-size flashlight when it needs to be. I've used it for nature study at night; it's fine for confronting a possum or deer thirty feet away, if the need arises. And I regularly use it when doing astronomy, on the lowest setting.

But wait. Don't astronomers need red penlights to preserve night vision? We'll get to that next.

MOCAP: Plastic caps for astronomy (and many other uses)

You probably never thought of plastic caps as basic tools of astronomy. But think about it. Don't we need lens caps in all sorts of oddball sizes, up to several inches across? Not to mention protective covers for the ends of screws and axles?

Let me introduce MOCAP, a company that not only makes these things, but also sells them in small quantities ("micropacks," typically $20 or $30 or so worth of an item).

In my limited experience, their RVC round vinyl caps stretch about 5%, which means they fit anything in a wide range of sizes, while their straight poly caps are more rigid (and flat-bottomed). Their unthreaded pipe caps have possibilities as lens caps for eyepieces, camera lenses, and telescopes.

The S series poly caps are red, which means they work well as night-vision covers for penlights. The S-9/16-RD1 is perfect for "astronomizing" the ThruNite Ti4T, while the S-1/2-RD1 is a slightly loose fit on more common penlights (a bit of tape will make it snug).

The advantage of a red cap, rather than a red penlight, is that I can pull it off if I suddenly need more light to shoo away an animal or deal with an emergency.


50 years of amateur astronomy

I have been an amateur astronomer for half a century. My first planned observing session, by myself, took place in October 1967, the month after my tenth birthday. The date is not recorded, but it was a cold, moonless night. Georgia did not have Daylight Saving Time at the time, so I probably went outside around 9 p.m., well after the end of twilight. Using my grandfather's 7×35 binoculars, which you see above, I was greatly impressed by a rich star field in what must have been central Cygnus, though I did not succeed in identifying any stars or constellations. I do remember consulting a planisphere and noting down, "Lacerta in zenith." Lacerta is the constellation just east of Cygnus.

This was also my first view of the stars from a dark country site, the home of my grandparents Covington on what is now Lans Lane (then Covington Road) about nine miles south of Millen, Georgia. The house is no longer standing; it was at latitude 32°41'30"N, longitude 81°59'13"W, according to Google Maps. I think the oldest of the farm buildings now on the site is my grandfather's garage.

Just now, I re-observed the same star field with the same binoculars, though not from a dark site (I hope to do that at Deerlick next week). To my delight, these binoculars focus to infinity with my myopic eyes, so I surely used them without glasses in 1967, when I was less myopic. They claim an 11-degree field, but the image is only sharp at the center, and even then, not extremely sharp — just the kind of thing that would call attention to Milky Way star clouds rather than stars.

The other objects in the picture tell more of the story. The planisphere is similar to the one I used that night, except that it didn't have the color overprinting. The book, Baker's Astronomy, was lent to me by my sixth-grade teacher, but I'm not sure whether this happened in the fall of 1967 or the following spring. (This is a copy I bought later, of course; I gave the borrowed one back!)

The way I got into astronomy was by reading ahead in the science textbook very early in the school year. That led to more reading about astronomy and to the observing session I've just described. But I was frustrated by not being able to identify constellations, and I put astronomy aside and pursued other interests until, in the spring of 1968, I noticed three stars in the evening sky and identified them as the belt of Orion. Then things fell into place quickly; I read every astronomy book I could get my hands on, savored a class trip to the VSC Planetarium in April and a lunar eclipse the next night, and started pursuing astronomy in earnest.


What is 6÷2(1+2) ?
A "puzzle" because PEMDAS is not the whole truth

A "mathematics puzzle" is going around in which people argue about the value of


and get either 1 or 9.

Everyone agrees that (1+2) is equivalent to (3). So we have:


and then the dispute begins.

Elementary schools teach the PEMDAS rule: Parentheses (which we've done), Exponentiations (which we don't have), Multiplications and Divisions, Additions and Subtractions. According to the PEMDAS rule, you do multiplications and divisions in the same step, left to right. Thus:

6÷2(3)  =  6 ÷ 2 × 3  =  (6 ÷ 2) × 3  =  9

But a lot of other people, including me, want to do the multiplication 2(3) before the division:

6÷2(3)  =  6 ÷ (2 × 3)  =  1

That is, we treat implicit multiplication (multiplication written without an ×) as grouping more tightly than ordinary multiplication.

And then the first crowd says, "That's wrong! That's not PEMDAS!"

Well... Several things need to be said.

(1) Mathematics isn't a computer language and doesn't have to follow the same rules of notation everywhere.

(2) It is indeed widespread, though not universal, practice to treat implicit multiplication as grouping more tightly.

And the really big one...

(3) Opportunities for testing this distinction are very rare! The reason is, people don't normally write formulas with alternating multiplications and divisions. They're very awkward! Instead, we group multiplications together, so that instead of

a × b ÷ c × d

we would write

(abd) / c

probably with a fraction bar rather than a slash. If working with polynomials (a major part of algebra), we would even eliminate the division by changing the value c to its reciprocal, so that only multiplications are used, and we just have abcd. The idea is that we see the factor structure of each term by seeing what is multiplied.

And that brings up another notational conundrum. When mathematicians type a slash for division, they are often thinking of a fraction bar, and they may intend for it to group less tightly than other multiplications and divisions. That is,


may be intended to represent



in which the multiplications are done first, because the fraction bar sets expressions apart just the way parentheses do. But we're not sure. A more careful typist would have written (ab)/(cd).

Bottom line? PEMDAS or no PEMDAS, when someone writes 1/2x, he or she almost certainly does not mean (1/2)x.

And an even more important point: This is not knowledge of mathematics. It's just knowledge of notation. A student with a talent for mathematics is the kind of person who can tell at a glance that 60 is divisible by 3 and 40 isn't. But knowing PEMDAS is just a matter of having memorized something that is traditional, arbirary, and only partly applicable!

Why people can't spell lens

In any online forum that has anything to do with optics, lens is one of the most misspelled words I ever encounter. Of the four forms

len lens lense lenses

only the two underlined ones are real words. But I see the others a lot, and even hear the first one spoken. What's going on?

Historical linguistics (the study of language change) provides the answer. Very simply, languages try to put words into familiar patterns. The word lens is the only English word with a singular ending in -ens, as far as I know, and it competes with two patterns that are much more common:

Pattern  Singular  Plural  
1lenslensesonly word in its class
2lenselenseslike sense/senses, expense/expenses, etc.
3lenlenslike pen/pens, den/dens, glen/glens, etc.

Because pattern 1 has only one word in it, the language tries to simplify itself by either putting lenses into pattern 2 and back-forming lense, or, a bit more violently, taking lens to be a plural in pattern 3 and back-forming a new singular len.

If languages are constantly regularizing themselves, why is there any irregularity in them? Because irregularity has two sources. In this case, lens is unlike other English words because it was borrowed, whole, from Latin, where -ens endings are common. (So was Homo sapiens, which also confuses people.) The other way irregularity arises is through sound change, so that a change in pronunciation breaks up a pattern or moves a word out of it.

I'm a stickler for correct spelling now, but I think that in the long run, pattern 2 (singular lense) is going to catch on. The only problem is that the pronunciation /lenz/ matches pattern 3 better than pattern 2; sense is pronounced /sens/. For all I know, the pronunciation will change, and lens will become /lens/. I've already heard devotees of pattern 3 say /len/.

Short update

I've been alternating periods of hard work and brief illness (cough, cough!) but will be back to the Daily Notebook, at least sporadically, in the coming days.


Happy birthday, Melody!

I want to wish Melody a happy birthday today (October 5). My best friend and the love of my life are one person, which works out very conveniently! Thank you, Melody, for almost 42 years, so far, of friendship and steadily increasing devotion.

As Melody recovers from her hip problem and from 1.5 years of being unable to walk, I find myself feeling the same anxieties as a parent whose child is growing up. Am I neglecting her if I don't follow her around and offer assistance continuously? Will she get something to eat if I don't bring it? Can she actually leave the house without me?

She is certainly enjoying being able to do her own driving, cooking, and so forth, and I am enjoying being a bit less pressed for time.

We used to be very young...

I've told the story of Melody's and my courtship in several Daily Notebook articles, but I've left out a key fact: we were younger than the average college students when we met.

Melody has an October birthday and was about 5 weeks below the minimum age for starting first grade, but they let her start anyhow. That made her the youngest person in her class every year.

I did better than that. I was 2 weeks below the minimum age when I started first grade, and then I was immediately promoted to second. Then, at the end of high school, I had no twelfth grade year at all, going straight to the University of Georgia and arriving perhaps two and a half years younger than the average freshman.

That is how we came to be 17 and 18, respectively, when we met at a recruiting event, even though I was already a third-year college student, and she was a high-school senior jointly enrolled at Gainesville Junior College.

And that is why we felt we could take our time. The wedding came nearly seven years after we met.


Periodic-error correction

Modern astronomy has spawned an interesting mechanical invention, periodic-error correction (PEC). When a motor needs to move precisely (to make a telescope track the stars), the irregularities of its gears can be recorded by a microprocessor, then played back as corrections to the motor speed each time the slowest gear goes around.

Below you see a crude but successful example. This is a graph of how well my Celestron CGEM mount tracked a star for about 8 minutes with and without PEC. The low-level fluctuation is the unsteadiness of the air. Pay attention to the blue graph; the red one is the other axis, which was not motor-driven.

It's possible to do at least twice this well. I made my recording by averaging three turns of the gear. I should average nine and do it in steadier air. Also, the CGEM is known to have some errors that are not periodic with the main gear (due to non-integer gear ratios). Still, not bad!


The Porsche 911 of vacuum cleaners?

About 25 years ago we bought a Miele "Red Star" canister vacuum cleaner, and about 20 years ago we bought another one, a "White Star," to use upstairs. Recently, I swapped them, so that the less-used one could take over the heavier duties, and then set out cleaning up the older one and doing some minor repairs.

These things are built to last forever. The only problems with the red Miele, after 25 years of heavy use, were that it smelled bad, the door to the tool compartment was loose, and one of the hinges of the main hood (which exposes the filter) was bent.

I was able to get another tool-compartment door on eBay for $40; it might actually have cost less directly from Miele, whose parts department was very helpful. The hinge repair was easier: the hinge (made of black plastic) resumed its normal shape as soon as I heated it. (The stresses in the plastic were relieved, and it went back to its cast shape.) I had been expecting to have to bend it.

Getting rid of the smell was harder. Obviously, a new bag and new HEPA filter were called for. But I also needed to clean every interior surface, and for that purpose I took it completely apart.

Hint: To get into a Miele canister, open the hoods, and then note that the switch assembly (black) is held in place by two screws and a hidden electrical connector. Remove those screws and gently pry it up, disengaging the connector. Then you can get to all of the screws for removing the top cover.

Hint #2: The main hood lifts off without tools; open it and then carefully push outward (or was it inward?) on it near the hinges.

Hint #3: There are videos on YouTube about how to overhaul a Miele.

The smell resided in a thin coating of finely powdered greasy dust all over the interior. To get rid of it, I used Febreze like cleaning fluid, spraying it on surfaces and then wiping it off with a rag, often several times. There were a couple of thick pads for the motor; I vacuumed them (with another vacuum cleaner!) and then applied Febreze and let it dry.

It worked.

While doing all this I concluded that Miele vacuum cleaners are remarkably well built and are classic designs that change little from year to year. Certainly, in 25 years we did not lose compatibility with any of today's accessories, and many of the repair parts are unchanged.

What's more, like Porsches and Volvos, they have a loyal following of technicians (professional and do-it-yourself) who appreciate the engineering and like to keep the machines running, and who have appeciable community spirit. I have gotten great help from eBay vendors of parts. Some of them are obviously in it because they appreciate the technology more than because they want to make money.

[Addendum:] I should add that the power head and the hose are not as durable as the canister; we've replaced both over the years. It's good that they're replaceable. They are also, for the most part, compatible across models, so our white one now has a power head that didn't come with it.

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