Why Math?

… sharing this from a discussion on learning math on Quora. I agree with the writer on most of it:

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I don’t think it’s actually a question of IQ. Anyone of average intelligence can understand mathematics provided that they learn in a sequential way and they follow a well-developed curriculum. I like to believe that mathematics is logical enough so that even the most difficult concepts can be grasped if it is explained by a patient teacher.

I’d like to posit a different question: why do people of average intelligence dislike (perhaps hate, fear, and despise) mathematics?

Here are 7 possible reasons:

  1. They had horrible teachers when they were little who humiliated them. Mathematics was weaponized by bitter people to bludgeon their student’s budding sense of identity.
  2. They missed learning essential skills in early grades which made it difficult to understand slightly more advanced mathematical steps at a higher grade level.
  3. They never got to a point where they saw the aesthetic nature of math and that nature itself appears to be entirely based on mathematical principles. (Once someone gets to this point, mathematics is as delightful as drawing, painting, or sculpting.)
  4. They resorted to memorizing formulas without understanding the underlying order of any mathematical idea.
  5. They were forced to do mathematics to pass an examination rather than introduced to it as a conceptual tool (probably the best one humans have ever invented or discovered.)
  6. They never wondered if humankind invented mathematics or if mathematics is actually the fabric of reality that astute human beings have observed and reflected on.
  7. They never marveled at the raw genius of someone like Srinivasa Ramanujan or how Issac Newton and and Gottfried Leibniz independently invented calculus during the mid 17th century. The beauty of these romantic stories about mathematics completely escapes them.

Prasad’s Home-Grown Drive Controller

Prasad D made a great mirror in our telescope-making workshop here at the Chevy Chase Community Center, and then proceeded to machine a wonderful Crayford focuser, from scratch, after I showed him how to use our 1944-era South Bend lathe. A very friendly fellow, he unfortunately (for us) moved to Philadelphia, which is not really close, but he’s kept up doing excellent ATM work.

As an example, on Tuesday last he brought in a brand new German equatorial drive and controller system that he had cobbled together from various parts. He replaced the original motors in the drive head with stepper motors, and then put together an Arduino board, a wireless communicator, and two stepper-motor controllers. All of the circuit is controlled from an app that he devised, on his Android device. We didn’t have any clear skies to try it out, but I could certainly see the motors slewing to various invisible objects such as the star Procyon and Messier Object 42..

Really first-rate job, and very nicely done! (He said he didn’t want it to look half-baked, and it doesn’t!)

Prasad asked me to “please give credit to the original creator of the electronics – Howard Dutton. He calls the system OnStep. It is based on Arduino Teensy3.2 microcomputer and it can be customized for any type of mount including Dobsonian. It is very easy to work with and your students at CCCC may find it interesting.”
I see a web page with lots of information: http://www.stellarjourney.com/index.php?r=site/equipment_onstep

I’ll post some still photos here and then upload two short videos to Youtube – which I cannot embed on this blog, but can only link to. If you click on the photos, you can see larger images.

 

 

Here is the first video that I took of his device in operation: https://youtu.be/Md06jD35CUg

Here is the link to a shorter video, not as detailed, that I took of the device:  https://youtu.be/bH_i58LhtiY

If you have problems viewing any of this, please let me know by leaving a comment. Thanks.

Quantifying Progress in the Fight Against Turned Down Edge

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By Guy Brandenburg

3/27/2017

I describe here an attempt to quantify progress (or lack thereof) in the removal of the classic, and dreaded, turned-down edge (TDE) present on a 16.5” Newtonian glass mirror blank that I have been trying to “figure” for some years.  The figuring process means changing a piece of glass that approximates a small section sliced out of a large hollow sphere, into a highly-accurate paraboloid — with the required level of accuracy being measured in nanometers.

 

Many amateur and professional telescope makers have maintained that you can only fix figuring errors if you can measure them. Not being able to get good, repeatable measurements of the TDE on my mirror, I had been sort of floundering, failing to get rid of the TDE even after YEARS of work (off and on; mostly off). So a decision was made to try to quantify things.

 

We recently had some success in matching computer-generated Ronchi images of theoretically-perfect mirrors with photos taken of works in progress, simply by cutting and pasting – which has been recommended by Mel Bartels in particular for quite some time. For the first time, I got the hang of it, and we were able to help a first-timer (Mike L) to figure a 10” plate glass f/5.4 mirror only ¾” thick to just about exactly ¼ lambda, according to our combined, repeated, careful measurements on a mirror that was cooled both by immersion in a room-temperature water bath and by sitting in a closet in the very same testing room for an entire weekend.

 

Prior to this experiment, I had been taking short videos of the entire mirror, moving the ronchi grating back and forth across the center of curvature. These videos reveal and record a lot of qualitative information about the mirror, including vocal commentary, but I found it impossible to transfer the images to my laptop for closer analysis until I got home, across town, which meant that the turn-around time after testing a mirror was much too long to be of any use. I had tried quite a large number of various strokes suggested by others, by our reading various ATM manuals, and by just thinking; but the very serious TDE on this (for me, relatively ambitious) project never seemed to get any better.

 

I simply gave up on imaging via video clips, since they were too hard to manipulate or measure on my phone, and which required too much bandwidth to send to my laptop until I got home. This time, I took Ronchi still-images on my cell phone, between 0.2 and 0.5 inches outside of the center of curvature.

guys 16 perfect

(My experience has been generally easier to discern defects in a Ronchigram when the lines curve outwards at the top and bottom, which would mean the test grating is  OUTSIDE the COC of a partly-parabolized mirror, as you see on the left in the black-and-white image above. However, when the lines curve inwards at the top and bottom, like the images in the center and to the right, then many serious defects remain hidden. quantify TDE

Procedure:

A standard 100 LPI grating from Willmann-Bell and a yellow LED were used, on an XYZ stage partly fabricated by me and placed exactly twice the focal length from the primary. Images were taken with an iPhone 6, shooting images zoomed in as much as possible. An attempt was made to have matching ronchigrams, i.e., with the same number of vertical lines showing.

 

(This was a weak point of the experiment. For one, it’s hard to hold cell phone steady enough, and an observer will notice that the images do NOT have exactly the same number of lines. That’s because I did not have a printout of the previous image right in front of me to make comparisons to. All that needs to be fixed in subsequent iterations. Also, other imaging devices need to be tried, as well.)

 

I was in fact able to email individual photograph frames to my laptop at the lab. After downloading the clearest images to my laptop, I used plain old MS Windows Paint to shrink and crop the useful portion of the picture, and then pasted the result into a Geometry software (Geometer.s Sketchpad, or GSP) that I was already familiar with. GSP was then used to draw a circle around the circumference of the image of the nearly-perfectly-circular glass disk, adjusting this as well as possible. This process automatically generated the center of the disk. Using that center, a second, and smaller, circle was drawn whose circumference was placed at the location along the ronchi lines where they appeared to begin to turn outwards. GSP was then  to measure directly the radii of the two circles and to compute their ratio.

 

A final ratio of 0.7, just to pick a number that is easy to compute, means that just about half of the area of the mirror is covered by a wide rolled-down edge, since the ratio of areas is equal to the square of the ratio of the respective radii, and 0.7 squared is 0.49, or 49%.

 

In the diagram above, the images go in chronological order but COUNTER-clockwise, from upper left (labeled #1), which was made in mid- or early March, through the next three images, all taken on March 22. In between each image, various strokes were employed in figuring sessions for anywhere between 15-20 minutes to attempt to fix the TDE. All the figuring sessions involved sub-diameter laps anywhere from 8 to 12 inches in diameter that had been warm-pressed upon the mirror. The strokes were both forward and back and incorporated enough of a ‘W’ stroke to cover the entire mirror, using cerium oxide on either tempered burgundy or Acculap pitch, depending. The edge of the tool was allowed to go up to the edge of the mirror, +/- maybe 5 mm. The goal was simply to wear down the glass in the center until it caught up with the amount that the edge had been worn down. None of the laps seemed to have full contact with the mirror out to the very edge; thus the end of the stroke was NOT at the edge of the mirror.

 

You will notice that these ratios, circled in green, seem to increase monotonically from 69% to 80%, which is gratifying: if this real, then the fraction of the mirror that is NOT covered by TDE has gone from about 47% to about 67%, as you can see here. (Note: in figure #1, the large circle was denoted circle AB, and the smaller circle was denoted circle CD. I know that points A and C are not identical, but they are rather close; that error will be fixed in subsequent iterations.)

However: the key question is: IS THIS REAL? Or am I merely fooling myself?

I don’t know yet.

I certainly hope it is real.

But it needs to be checked with subsequent investigation.

My attempt at limiting my own subjectivity or wishful thinking was to try to draw the circles at the place where the more-or-less vertical lines began turning outwards. Hopefully that location really corresponded to the place where the turned/rolled edge began. However, it is entirely possible that the precise apparent location of the beginning of the TDE very much depends on exactly how many lines appear in the Ronchigram, thus, precisely how far from the COC the grating is located.

Unfortunately, often times I have to dismantle the entire apparatus, because we have to close up shop for the night, or somebody else needs to use the tester on another mirror. Thus, it is nearly impossible to ensure that the measurement apparatus remains undisturbed.

My next steps, I think, are these:

  1. Have a separate, and very simple ronchi apparatus that just consists of a grating and a light.
  2. Have previous images right in front of me as I prepare to take the next Ronchigrams, so that I can match the number of lines visible.
  3. Perhaps I should take a series of said standardized ronchigrams both inside and outside of COC with, say, 5 lines visible. I should also take some ronchigrams that might accentuate and expose any possible astigmatism; that is, very close to the COC. Any Ronchi lines that resemble the letters S, Z, J, U, or N would be very bad news.
  4. Attempt to attach a cheap video camera with built-in LED, Ronchi grating, and a suitable lens to make steadier images free from hand wobbles.

I would like to thank Isaac and Elias Applebaum for their diligent and noted explorations in solving a similar question relating to fixing or preventing TDE. That STEM project won them a number of well-deserved awards.

 

 

What a Great Night!

Just got back from an exciting astro expedition to Hopewell Observatory with one of the other members. Great fun!

Anybody living on the East Coast in March 2018 has just lived through a very strong, multi-day gale. The same weather system brought snow and flooding to the northeast, and here in the DC-Mar-Va area, it was cut off power to many (including my mother-in law) and caused almost all local school districts to close — even the Federal Government! Two of my immediate neighbors in DC had serious roof damage.

Today, Sunday, Paul M and I decided the wind had calmed enough, and the sky was clear enough, for an expedition to go up and observe. We both figured there was a good chance the road up to the observatory would be blocked by trees, and it turns out that we were right. My chainsaw was getting repaired – long story, something I couldn’t fix on my own – so I brought along work gloves, a nice sharp axe, loppers, and a 3-foot bowsaw. We used all of them. There were two fairly large dead trees that had fallen across the road, and we were able to cut them up and push them out of the way.

However, there was a large and very dangerous ‘widow-maker’ tree (two images above) that had fallen across the road, but it was NOT on the ground. Instead, was solidly hung up on the thick telecommunications line at about a thirty-degree angle to the ground. The power lines above it didn’t seem to be touched. You could easily walk under the trunk, if you dared (and we did), and you probably could drive under it, but of course the motion of the car just might be enough to make it crack in half and crush some unlucky car and its driver. Or maybe it might make the phone line shake a bit …

No thanks.

So, we didn’t drive under.

I called the emergency phone for the cell phone tower (whose access road we share) to alert them that the road was blocked and could only be cleared by a professional. I also attempted to call a phone company via 611, without much success — after a long wait, the person at the other end eventually asked me for the code to my account before they would forward me to somebody who could take care of it. Very weird and confusing. What account? What code? My bank account? No way. We will both call tomorrow. Paul says he knows some lawyers at Verizon, whose line he thinks it is.

But then: how were we going to turn the cars around? It’s a very narrow road, with rocks and trees on one side. The other side has sort of a ravine and yet more trees. Paul realized before I did that we had to help each other and give directions in the darkness to the other person, or else we would have to back up all the way to the gate! Turning around took about four maneuvers, per car, in the dark, with the other person (armed with astronomer’s headlamp, of course) yelling directions on when to turn, how much to go forward, when to stop backing up, and so on. Success – no injuries! We both got our cars turned around, closed them up, got our cutting tools, gloves and hats, and then hiked the rest of the way up, south and along the ridge and past the big cell phone tower, to the Observatory buildings themselves, moving and cutting trees as we went.

As we were clearing the roadway and walking up the ridge, we peered to the west to try to find Venus and Mercury, which had heard were now evening planets again. It wasn’t easy, because we were looking through LOTS of trees, in the direction of a beautiful multi-color, clear-sky sunset featuring a bright orange line above the ridge to our west. Winter trees might not have any leaves, but they still make the search for sunset planets rather tough. Even if you hold perfectly still, one instant you see a flash that’s maybe a planet, or maybe an airplane, and then the branches (which are moving in the breeze, naturally) hide it again. So what was it? Paul’s planetarium smartphone app confirmed he saw Venus. If the trees weren’t there, I think we also would have seen Mercury, judging by Geoff Chester’s photo put out on the NOVAC email list. I think I saw one planet.

In any case, everything at the observatory was just fine – no tree damage on anything, thanks to our prior pruning efforts. The Ealing mount and its three main telescopes all worked well, and the sky and stars were gorgeous both to the naked eye and through the scopes. Orion the Hunter, along with the Big Dog and the Rabbit were right in front of us (to the south) and Auriga the Charioteer was right above us. Pleiades (or the Subaru) was off high in the west. Definitely the clearest night I’ve had since my visit to Wyoming for the solar eclipse last August, or to Spruce Knob WV for the Almost Heaven Star Party the month after that.

Paul said that he and his daughter had been learning the proper names of all the stars in the constellation Orion, such as Mintaka, Alnilam, and Alnitak. As with many other star names, all those names are Arabic, a language that I’ve been studying for a while now [but am not good at. So complicated!] Mintaka and Alnitak are essentially the same Arabic word.

After we got the scopes working, Paul suggested checking out Rigel, the bright ‘leg’ of Orion, because it supposedly had a companion star. {Rajul means “leg”} We looked, and after changing the various eyepieces and magnifications, we both agreed that Rigel definitely does have a little buddy.

I had just read in Sky & Telescope that Aristotle (from ancient Greece) may have given the first written account of what we now call an “open cluster” in the constellation Canis Major (Big Dog – that’s Latin, which I studied in grades 7 – 12) called Messier-41, only a couple of degrees south of Sirius, the brightest star in the sky. A passage in a book allegedly written by Aristotle (roughly 230 BC) seems to indicate that he could see this object with averted vision. (He was trying to establish that it was a fuzzy patch in the sky that was most definitely NOT a comet, just like Charles Messier was doing almost exactly two thousand years later!)

M41 was quite attractive. But no, we didn’t then look at M42. Been there, done that many times before. And no, what you see with a telescope does not have all those pretty colors that you see in a photograph.

Instead, we looked on a multi-sheet star atlas (that stays in the observatory) near M41 and found three other open clusters, all really beautiful. We first found M38 and thought that in the C-14 and 6″ Jaegers, it looked very anthropoid or like an angry insect, if you allowed your mind to connect the beautiful dots of light on the black background. In the shorter 5″ refractor made by Jerry Short, it looked like a sprinkling of diamond dust. This cluster must have been formed rather recently. We then found M36, which was much less rich, but still quite pretty. Lastly, we found M37, another open cluster, which has a very bright yellow star near the center, against background of much fainter stars. It seemed to me that those other stars might be partly obscured by a large and somewhat translucent cloud of dust. We saw a web of very opaque dust lanes, which we confirmed by readings on the Web. Really, really beautiful. But I’m glad we don’t live there: too dangerous. Some of the stars are in fact red giants, we read.

Then we looked straight overhead, in the constellation Auriga. We decided to bypass the electronics and have Paul aim the telescope, using the Telrad 1-power finderscope, at one of the fuzzy patches that he saw there. He did, and my notes indicate that we eventually figured out that he found Messier-46 (yet another open cluster) with his naked eye! Very rich cluster, I think, and we even found the fan-shaped planetary nebula inside!

At this point we were getting seriously cold so we moved over just a little, using the instruments, to find M47, again, a very pretty open cluster.

Realizing that the cold and fatigue makes you do really stupid things, and that we were out in the woods with no way to drive up here in case of a problem, we were very careful about making sure we were doing the closing up procedures properly and read the checklist at the door to each other, to make sure we didn’t forget anything.

On the walk back, we saw the Moon coming up all yellowish-orange, with the top of its ‘head’ seemingly cut off. When it got a bit higher, it became more silver-colored and less distorted, but still beautiful.

I really thought all of those open clusters were gorgeous in their own right, and I think it would be an excellent idea to make photographs of them, but perhaps black dots on white paper, and give them to young folks, and ask them to connect the dots, in whatever way they feel like doing. What sorts of interesting drawings would twenty-five students come up with?

I am not sure which of our various telescopes would do the best job at making astro images. I have a CCD camera (SBIG ST-2000XM), with a filter wheel. What about just making it a one-shot monochromatic black and white image? I also have a Canon EOS Revel XSI (aka 450D, I think). Compare and contrast… The CCD is really heavy, the Canon quite light. I also have a telephoto lens for the Canon, which means that I have essentially four telescopes to choose from (but not a big budget!). One problem with the C-14 and my cameras is that the field of view is tiny: you can only take images of very small bits of what you can see in the eyepiece with your naked eye. This means you would need to make a mosaic of numerous pictures.

In any case, no imaging last night! Not only did I not feel like hauling all that equipment for a quarter of a mile, after all that chopping, sawing, and shoving trees, it turns out I had left my laptop home in the first place. D’oh!

I had previously found every single one of these open clusters when I made my way through the entire Messier list of over 100 objects, with my various home-made telescopes, which had apertures up to 12.5 inches. However, I don’t think I had ever seen them look so beautiful before! Was it the amazing clarity of the night, or the adventure, or the company? I don’t know!

But this was a very fun adventure, and this photography project – attempting to make decent images of these six open clusters – promises to be quite interesting!

 

 

 

 

 

A 6″ Dob for Young Relatives

I just finished a 6″ f/8 Dobsonian telescope as a gift for my great-nephews, one of whom I discovered is VERY interested in astronomy and happens to live in a place with pretty dark skies – about the middle of Maryland’s Eastern Shore. The mirror is excellent, and the mechanical parts all work very well — in my opinion. Let’s see what the recipients think.

I finally succeeded in putting in the mirror yesterday afternoon in a very stiff and cold wind right outside my house, estimating where the mirror should go by aiming at a distant chimney. This takes patience because it’s trial-and-error, no matter how much you calculate beforehand!

Later that evening, after the nearly-full Blue Moon came over the trees at DC’s Chevy Chase Community Center where we have our telescope-making classes, fellow ATMer and all-around interesting person Jim Kaiser helped me collimate it by pointing the scope at the illuminated curtains in the windows of the CCCC. We then verified that the Moon actually did come to a focus with the eyepiece nearly all the way screwed in. If you want to focus on closer things than the Moon or galaxies, you need to screw the eyepiece out towards you, the observer in the cheap but effective helical focuser that was lying around the shop.

This scope incorporates a couple of innovations by me, and a bit of artistic whimsy.

First small innovation: I made the secondary diagonal mirror holder so that no tools are needed at all: you just rotate the part holding the elliptical mirror and turn a little thumbscrew to collimate it quickly and easily, while you watch. Here is a sketch of how I made it.new type of secondary holder

 

Second innovation can be seen near my right hand (to your left) atop the cradle: two 1/4″-20 machine screws with simple homemade knobs on top, going through threaded inserts (T-nuts would work too), which push against a piece of lumber in the shape of prism with an isosceles right triangle at each end. I call this the tube brake, which can be applied or released quite easily, whenever needed. Small springs (almost impossible to see in this photo) pull this brake up against the corner of the tube, while the machine screws press it down. If you want to change the position of the eyepiece because a taller or shorter person has arrived, no problem. A few CCW turns of the wooden knobs releases the brake, you rotate the tube to the desired position, and then you lock it down again with a few clockwise turns. If you add or remove a heavy eyepiece or a finder or whatever, same procedure, except this time you can slide the scope up and down inside the cradle.

The artistic whimsy is partly seen in a photo Jim took of me after we got it collimated but before we rushed back inside: lots of colors, thanks to several tons of paint cans salvaged by fellow ATMer Bill Rohrer from being thrown away by a third party who lost his warehouse lease, and also because smurf blue is the favorite color of one of the boys. The altitude bearing is made out of the Corian countertop that my wife and I got rid of a few months ago when we had our kitchen remodeled. (30 years ago we did it ourselves, mostly. This time we hired professionals. They are SOO much faster and better at this than us!)

So that my young relatives can keep this thing looking good, they also get four or five quart or pint cans of paint – the ones I used on the scope. Free, of course. The more we get rid of put to use, the better. They can repaint anything that gets scratched, you see?IMG_9416

You can also see some wood-cutting fun above and below. This retired geometry teacher had a lot of fun figuring out how to lay out and cut out stars with 5, 6, and 7 points, as well as a crescent moon and a representation of Saturn seen with its rings edge-on. I guess you could show Saturn’s rings a 30 to 45 degrees to the viewer, if you instead carved it out of solid wood or did wood burning, but I just had a hand-held jigsaw and a Dremel knockoff. And plus, this is supposed to be a scope that is USED rather than just admired for its artsy parts.

I designed what I wanted onto two sheets of paper and then taped them to the plywood. This worked, but it wasn’t the most wonderful plywood, so on many of the pull strokes, the wood splintered a bit. So that side got to face inside.. Painting all those little nooks and crannies was tough!

design artsy astro cutouts

(The purpose of the cut-outs was simply to make the telescope lighter. It’s got a very heavy and sturdy base. Each square inch of plywood removed saves about 7 grams. Also, more holes means more hand-holds!)

 

 

Under-corrected Commercial Mirrors

I have tested three different Newtonian mirrors from 12″ to 16″ diameter over the past few years that were labeled as being by Meade. One had been refigured by someone else before I tested it.

Each of the mirrors initially looked nice and smooth, and the Ronchigrams looked pretty close to the theoretically-perfect image generated by RonWin or similar software.

However, when I did a zonal test, in every single case, the mirror turned out to be seriously under-corrected, in some cases by about 50%. In other words, if a perfect mirror should have the outer zone (say the outer inch) to “null out” at 0.236″ from the location where the central zone nulls out, then the mirrors I tested might null out at only 0.118″ instead.

I took care to repeat the measurements several times in each of these cases, and in one case the owner also took a set of readings; his and mine agreed pretty closely.

I don’t know if it’s my skills at reading Foucault/Couder shadow zones that are suspect, or if I’m correct. Anybody else have similar or opposite stories or experiences?

Make your own telescope!

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This weekend, I’m hosting a small workshop at the Community Museum in Gaithersburg, MD, where interested persons from 8 to 88 years of age can make their own telescope in an hour or two. We will be using surplus but high-quality achromatic primary doublet lenses as well as inexpensive eyepieces, along with PVC tubing and some really cool tripods to hold it steady.  We will some basic optics experiments to help explain how these gizmos work, and will have spray paint and colored tape to decorate the tubes.

If you are interested, here is the necessary information:

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Gaithersburg Community Museum's photo.
JAN21

Make Your Own Telescope

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Make your own refracting telescope in just an afternoon! This workshop will be led by veteran DC amateur telescope-maker Guy Brandenburg. He will show you how to make a small functioning telescope that can either be held in your hands or mounted on almost any camera tripod. All of the materials needed will be provided, and no experience is necessary. This workshop is open to anyone from the ages of 9 through 99, but a parent would need to accompany any child from 9 – 11. You might get a little dirty, so don’t wear your best clothing!

You will also see how various types of telescopes such as reflectors, refractors, and catadioptrics are put together and operate, using actual examples, including the type made and used by Galileo around 1609.

$30 City of Gaithersburg residents/$35 non-residents. Space is limited to 15 so pre-registration is required. To register go to RecXpress at https://online.activenetwork.com/gaithersburg/Start/Start.asp. It’s activity #49690.

For more information or if you have trouble registering call the museum at 301-258-6160 or museum@gaithersburgmd.gov

 

Is This A Sketch of the Evolution of Galaxies Over Time?

The answer is, of course, no. Though they do bear some resemblance to the amazing astronomy graphics you come across these days at websites like Astronomy Picture of the Day, or graphs that astronomers show to their peers, these are merely wonderful doodles made by a fellow-volunteer at the registration desk at last week’s American Astronomical Society at the Gaylord convention center at the National Harbor in Oxon Hill, MD. Sorry, I have forgotten his name and I also didn’t get permission to post. (He was an undergrad or grad student but I forget where.) So I definitely need to pay him royalties if he ever tracks me down.

(He won’t get very rich though: even 12% or 37.5% or 99.99% of zero income from this blog is still zero…)

All of us volunteers and paid staff were very impressed by his doodles.

I’m sure that the reaction of a group of Freudian psychoanalysts would have been quite different.