There is a pretty well-known paradox which goes something like this:
You hear that the Smith and Jones families each have two children.
You are told that the older Smith child is a girl, and that at least one of the Jones children is a girl.
Assuming that boys and girls are equally likely to be born (I know this is not quite true, but let’s pretend) in any given pregnancy, what are the chances that the Smiths have two girls? How about the Joneses?
Most people would say that those probabilities are equal: 50% in both cases.
But they are not. In fact, it is much less likely that the Joneses have two daughters!
Here is why:
In any family with two children, there is an older sibling and a younger one.
In the Smith family, you know that the older child is a girl, but you know nothing about their younger child, so the younger one is equally likely to be male or female. So the chances that the Smiths have two daughters is indeed 1/2, or 50%.
In the Jones family, all we know is that there is at least one girl. Let’s look at a diagram that shows all of the equally-likely possibilities in any family with two children:
With the Smith family, we can rule out cases 1 and 2, leaving us cases 3 and 4.
However, with the Jones family, we can only rule out case number 1. Cases 2, 3, and 4 — which are all equally likely — are all possible outcomes for the Joneses. Notice that only in case number 4 do the Jones have two daughters. So with the Joneses, the chances that there are two daughters is only 1 in 3, or 33.3%.
Several people have helped me with this applied geometry problem, but the person who actually took the time to check my steps and point out my error was an amazing 7th grade math student I know.
It involves optical testing for the making of telescope mirrors, which is something I find fascinating, as you may have guessed. Towards the end of this very long post, you can see the corrections, if you like.
Optics themselves are amazingly mysterious. Is light a wave, or a particle, or both? Why can nothing go faster than light? We forget that humans have only very recently discovered and made use of the vast majority of the electromagnetic spectrum that is invisible to our eyes.
But enough on that. At the telescope-making workshop here in DC, I want folks to be able to make the best ordinary, parabolized, and coated mirrors possible with the least amount of hassle possible and at the lowest possible cost. Purchasing high-precision, very expensive commercial interferometers to measure the surface of the mirror is out of the question, but it turns out that very inexpensive methods have been developed for doing that – at least on Newtonian telescopes.
Tom Crone, a friend of mine who is also a fellow amateur astronomer and telescope maker, wondered how on earth we can report mirror profiles as being within a few tens of nanometers of a perfect paraboloid with such simple devices as a classic Foucault knife-edge test.
He told me his computations suggested to him that the best we could do is get it to within a few tenths of a millimeter at best, which is four orders of magnitude less precise!
I assured him that there was something in the Foucault test which produced this ten-thousand-fold increase in accuracy, but allowed that I had never tried to do the complete calculation myself. I do not recall the exact words of our several short conversations on this, but I felt that I needed to accept this as a challenge.
When I did the calculations which follow, I found, to my surprise, that one of the formulas I had been taught and had read about in many telescope-making manuals, was actually not exact, and that the one I had been told was inherently less accurate, was, in fact, perfectly correct! Alan Tarica sent me an article from 1902supposedly explaining the derivation of a nice Foucault formula, but the author skipped a few bunch of important steps, and I don’t get anything like his results. it took me a lot of work, and help from this rising 8th grader, to find and fix my algebra errors. I now agree with the results of the author , T.H.Hussey.
I am embarrassed glad to say that even after several weeks of pretty hard work, an exact, correct formula for one of the commonly used methods for measuring ‘longitudinal aberration’ still eludes me. was pointed out to me by a student who took the time to Let’s see if anybody can follow my work and helped me out on the second method.
But first, a little background information.
Isaac Newton and Leon Foucault were right: a parabolic mirror is the easiest and cheapest way to make a high-quality telescope.
If you build or buy a Newtonian scope, especially on an easy-to-build Dobsonian mount, you will get the most high-quality photons for the money and effort spent, if you compare this type with any other type of optics at the same diameter. (Optical designs like 8-inch triplet apochromats or Ritchey-Chrétiens, or Maksutovs, or modern Schmidt-Cassegrains can cost many thousands of dollars, versus a few hundred at most for a decent 8″ diameter Newtonian).
With a Newtonian, you don’t need special types of optical glass whose indices of refraction and dispersion, and even chemical composition, must be known to many decimal places. The glass can even have bubbles and striations, or not even be transparent at all! Any telescope that only has mirrors, like a Newtonian, will have no chromatic aberration (ie, you don’t see rainbows around bright stars) because there is no refraction – except for inside your eyepieces and in your eyeball. All wavelengths of light reflect exactly the same –but they bend (refract) through glass or other materials at different angles depending on the wavelength.
Another advantage for Newtonians: you don’t need to grind and polish the radii of curvature of your two or three pieces of exotic glass to exceedingly strict tolerances. As long as you end up with a nice parabolic figure, it really doesn’t matter if your focal length ends up being a few centimeters or inches longer or shorter than you had originally planned. Also: there is only one curved mirror surface and one flat one, so you don’t need to make certain that the four or more optical axes of your mirrors and/or lenses are all perfectly parallel and perfectly concentric. Good collimation of the primary and secondary mirrors to the eyepiece helps with any scope, but it’s not nearly as critical in a Newtonian, and getting them to line up if they get knocked out of whack is also much easier to perform.
With a Newtonian, you only need to get one surface correct. That surface needs to be a paraboloid, not a section of a sphere. (Some telescopes require elliptical surfaces, or hyperbolic or spherical ones, or even more exotic geometries. A perfect sphere is the easiest surface to make, by the way.)
In the 1850’s, Leon Foucault showed how to ‘figure’ a curved piece of glass into a sufficiently perfect paraboloid and then to cover it with a thin, removable layer of extremely reflective silver. The methods that telescope makers use today to make sure that the surface is indeed a paraboloid are variations and improvements on Foucault’s methods, which you can read for yourself in my translation.
Jim Crowley performing a Foucault test
It turns out that the parabolic shape does need to be very, very accurate. In fact, over the entire surface of the mirror, other than scratches and particles of dust, there should be no areas that differ from each other and from the prescribed geometric shape by more than about one-tenth of a wavelength of green light (which I will call lambda for short), because otherwise, instead of a sharp image, you just receive a blur, because the high points on the sine waves of the light coming to you would tend to get canceled out by the low points.
Huh?
Let me try to explain. In my illustrations below, I draw two sine waves (one red, one green) that have the same exact frequency and wavelength (namely, two times pi) and the same amplitude, namely 3. They are almost perfectly in phase. Their sum is the dark blue wave. In diagram A, notice that the dark blue wave has an amplitude of six – twice as much as either the red or green sine wave. This means the blue and green waves added constructively.
Next, in diagram B, I draw the red and green waves being out of phase by one-tenth of a wave (0.10 lambda) , and then in diagram C they are ‘off’ by ¼ of a wave (0.25 lambda). You will notice that in the diagrams B and C, the dark blue wave (the sum of the other two) isn’t as tall as it was in diagram A, but it’s still taller than either the red or green one.
One-quarter wave ‘off’ is considered the maximum amount of offset allowed. Here is what happens if the amount of offset gets larger than 1/4:
In diagram D, the red and green curves differ by 1/3 of a wave (~0.33 lambda), and you notice that the blue wave (which is the sum of the other two) is exactly as tall as the red and green waves, which is not good.
Diagram E shows what happens is what happens when the waves are 2/5 (0.40 lambda) out of phase – the blue curve, the sum of the other two, now has a smaller amplitude than its components!
And finally, if the two curves differ by ½ of a wave (0.5 lambda) as in diagram F, then the green and red sine curves cancel out completely – the dark blue curve has become the x-axis, which means that you would only see a blur instead of a star or a planet. This is known as destructive interference, and it’s not what you want in your telescope!
But how on earth do we achieve such accuracy — one-tenth of the wavelength of visible light (λ/10) over an entire surface? And if we do, what does it mean, physically? And why one-tenth λ on the surface of the mirror, when ¼ λ looked pretty decent? For that last question, the reason is that when light bounces off a mirror, any deviations are multiplied by 2. So lambda – about 55 nanometers or 5.5×10^(-8) m- is the maximum allowable depth or height of a bump or a hollow across the entire width of the mirror. That’s really small! How small? Really insanely small.
Let’s try to visualize this by enlarging the mirror. At our mirror shop, we generally help folks work on mirrors whose diameters are anywhere from 11 cm (4 ¼ inches) to 45 cm (18 inches) across. Suppose we could magically enlarge an 8” (20 cm) mirror and blow it up so that it has the same diameter as the original 10-mile (16 km) square surveyed in 1790 by the Ellicott brothers and Benjamin Banneker for the 1790 Federal City. (If you didn’t know, the part on the eastern bank of the Potomac became the District of Columbia, and the part on the western bank was given back to Virginia back in 1847. That explains why Washington DC is no longer shaped like a nice rhombus/diamond/square.)
So imagine a whole lot of earth-moving equipment making a large parabolic dish where DC used to be, a bit like the Arecibo radio telescope, but about 50 times the diameter, and with a parabolic shape, unlike the spherical one that Arecibo was built with.
(Technical detail: since Arecibo was so big, there was no way to physically steer it around at desired targets in the sky. Since they couldn’t steer it, then a parabolic mirror would be useless except for directly overhead. However, a spherical mirror does NOT have a single focal point. So the scope has a movable antenna (or ‘horn’) which can move around to a variety of more-or-less focal points, which enabled them to aim the whole device a bit off to the side, so they can ‘track’ an object for about 40 minutes, which means that it can aim at targets around 5 degrees in any direction from directly overhead, but the resolution was probably not as good as it would have been if it had a fully steerable, parabolic dish. See the following diagrams comparing focal locations for spherical mirrors vs parabolic mirrors. Note that the spherical mirror has a wide range of focal locations, but the parabolic mirror has exactly one focal point.)
I’ll use the metric system because the math is easier. In enlarging a 20 cm (or 0.20 m) mirror all the way to 16 km (which is 16 000 m), one is multiplying 80,000. So if we take the 5.5×10-8 m accuracy and multiply it by eighty thousand you get 44 x 10-4 m, which means 4.4 millimeters. So, if our imaginary, ginormous 16-kilometer-wide dish was as accurate, to scale, as any ordinary home-made or commercial Newtonian mirror, then none of the bumps or valleys would be more than 4.4 millimeters too deep or too high. For comparison, an ordinary pencil is about 6.8 millimeters thick.
Wow!
So that’s the claim, but now let’s verify this mathematically.
I claim that such a 3-dimensional paraboloid, like the radio dish in the picture below, can be represented by the equation
where f represents the focal length. (For simplicity, I have put the vertex of the paraboloid at the origin, which I have called A. I have decided to make the x-axis (green, pointing to our right) be the optical and geometric axis of the mirror. The positive z-axis (also green) is pointed towards our lower left, and the y-axis (again, green) is the vertical one. The focal point is somewhere on the x-axis, near the detector; let’s pretend it’s at the red dot that I labeled as Focus.)
You may be wondering where that immediately previous formula came from. Here is an explanation:
Let us define a paraboloid as the set (or locus) of all points in 3-D space that are equidistant from a given plane and a given focal point, whose coordinates I will arbitrarily call (f, 0, 0). (When deciding on a mirror or radio dish or reflector on a searchlight, you can make the focal length anything you want.)
To make it simple, the plane in question will be on the opposite side of the origin; its equation is x = -f. We will pick some random point G anywhere on the surface of the parabolic dish antenna and call its coordinates (x, y, z). We will see what equation these conditions create. We then drop a perpendicular from G towards the plane with equation x = -f. Where this perpendicular hits the plane, we will call point H, whose coordinates are (-f, y, z). We need for distance GH (from the point to the plane) to equal distance from G to the Focus. Distance GH is easy: it’s just f + x. To find distance between G and Focus, I will use the 3-D distance formula:
Which, after substituting, becomes
To get rid of the radical sign, I will equate those two quantities, because FG = GH, omit the zeroes, and square both sides. I then get
Multiplying out both sides, we get
Canceling equal stuff on both sides, I get
Adding 2fx to both sides, and dividing both sides by 4f, I then get
However, 3 dimensions is harder than 2 dimensions, and two dimensions will work just fine for right now. Let us just consider a slice through this paraboloid via the x-y plane, as you see below: a 2-dimensional cross-section of the 3-dimensional paraboloid, sliced through the vertex of the paraboloid, which you recall is at the origin. We can ignore the z values, because they will all be zero, so the equation for the blue parabola is
or, if you solve it for y, you get
There is a circle with almost the same curvature as the paraboloid; its center, labeled CoC (for ‘Center of Curvature’) is exactly twice as far from the origin as the focal point. You can just barely see a green dotted curve representing that circle, towards the top of the diagram, just to the right of the blue paraboloid. center of the circle (and sphere). Its radius is 2f, which obviously depends on the location of the Focus.
D is a random point on that parabola, much like point G was earlier, and D’ being precisely on the opposite side of the optical axis. The great thing about parabolic mirrors is that every single incoming light ray coming into the paraboloid that is parallel to the axis will reflect towards the Focus, as we saw earlier. Or else, if you want to make a lamp or searchlight, and you place a light source at the focus, then all of the light that comes from it that bounces off of the mirror will be reflected out in a parallel beam that does not spread out.
In my diagram, you can see a very thin line, parallel to the x-axis, coming in from a distant star (meaning, effectively at infinity), bouncing off the parabola, and then hitting the Focus.
I also drew two red, dashed lines that are tangent to the paraboloid at point D and D’. I am calling the y-coordinate of point D as h (D has y-coordinate -h)and the x-coordinate of either one is
I used basic calculus to work out the slope of the red, dashed tangent line ID. (Quick reminder, if you forgot: in the very first part of most calculus classes, students learn that the derivative, or slope, of any function such as this:
is given by this:
So for the parabola with equation
the slope can be found for any value of x by plugging that value into the equation
Since
the exponent b is one-half. Therefore, the slope is going to be
which simplifies to
Now we need to plug in the x coordinate of point D, namely
we then get that the slope is
To find the equation of the tangent line, I used the point-slope formula y – y1=m(x – x1). ; plugging in my known values, I got the result
To find where this hits the y-axis, I substituted 0 for x, and got the result that the tangent line hits the y-axis at the point (0, h/2) — which I labeled as I — or one-half of the distance from the vertex (or origin) to the ‘height’ of the zone, or ring, being measured.
Line DW is constructed to be perpendicular to that tangent, so any beam of light coming from W that hits the parabola at point D will be reflected back upon itself. Perpendicular lines have slopes equal to the negative reciprocal of the other. Since the tangent has slope 2f/h, then line DW has slope -h/(2f).
Plugging in the known values into the point-slope formula, the equation for DW is therefore
Here, I am interested in the value of x when y = 0. Substituting, re-arranging, and solving for x, I get
Recall that point C is precisely 2f units from the origin, which means that the perpendicular line DW hits the x axis at a point that is the same distance from the center of curvature CoC as the point D is from the y-axis!
Or, in other words, CW = AT = DE. This means: if you are testing a parabolic mirror with a moving light source at point W, then a beam of light from W that is aimed at point D on the paraboloid will come right back to W, and the longitudinal readings of distance will follow the rule h2/(4f), where h is the radius of the zone, or ring, that you are measuring. Other locations on the mirror which do not lie in that ring will not have that property. This then is the derivation of the formula I was taught over 30 years ago by Jerry Schnall, and found in many books on telescope making – namely that for a moving light source, since R=2f,
where LA means ‘longitudinal aberration and the capital R is the radius of curvature of the mirror, or twice the focal length. So that’s exactly the same as what I computed.
HOWEVER, this formula [ LA=h^2/(2R) ] does not work at all if your light source is fixed at point C, the center of curvature of the green, reference sphere. In the old days, before the invention of LEDs, the light sources were fairly large and rather hot, so it was easier to make them stationary, and the user would move the knife-edge back and forth, but not the light source. The formula I was given for this arrangement by my mentor Jerry Schnall, and which is also given in numerous sources on telescope making was this:
that is, exactly twice as much as for a moving light source. I discovered to my surprise that this is not correct, but it took me a while to figure this out. I originally wrote the following:
But now I can confirm this, thanks in part to two of my very mathematically inclined 8th grade geometry students. Here goes, as corrected:
If one is using a fixed light source located at the center of curvature C, and a moving knife-edge, located at point E, the the rays of light that hit the same point D will NOT bounce straight back, because they don’t hit the tangent line at precisely 90 degrees. Instead, the angle of incidence CDW will equal the angle of reflection, namely WDE. I used Geometer’s sketchpad to construct line DE by asking the software to reflect line CD over the line DW.
However, calculating an algebraic expression for the x-coordinate of point E was surprisingly complicated. See if you can follow along!
To find the x-coordinate of E, I will employ the tangent of angle TDE.
To make the computations easier, I will draw a couple of simplified diagrams that keep the essentials.
I also tried other approaches, and also got answers that made no sense. It looks like the formula in the 1902 article is correct, but I have not been able to confirm it.
I suspect I made a very stupid and obvious algebra mistake that anybody who has made it through pre-calculus can easily find and point out to me, but I have had no luck in finding it so far. I would love for someone did to point it out to me.
People have long wondered why flying insects can be seen spiraling around light sources at night. Among other suggestions was that the critters were used to navigating by the Moon, and got confused, or that they were seeking heat.
An ingenious new study shows that the navigation idea is not completely wrong, but the insects instead use sky glow, even at night, as a major clue for how to orient themselves: by keeping their dorsal (back) to a point or diffuse light source, for millions of years, then they would keep their legs pointed down and they would fly the way they want.
However, these researchers found that if they placed a light bulb in roughly the center of an otherwise darkened, enclosed space inside a tent with flying insects, then most (but not all) species of nocturnal insects flying above or to the side of the light tended to orient their bodies so that their dorsal side was towards the light— so that they were flying sideways or upside down! Thus disoriented, they would flitter around, confused as to which way was up.
This also explains why it is so easy to catch nocturnal flying insects by shining a bright light onto a sheet or blanket laid on the ground: convinced by hundreds of millions of years that “light = up”, a large fraction of the critters fly **upside down** towards the lighted surface and careen onto it, out of control.
Caution: This study has not yet been replicated or peer-reviewed, but if it holds up, then it unveils a very simple and inexpensive fix for both ever-worsening light pollution and the collapse of our global insect populations: simply put shielding around ALL exterior light fixtures at night, so that NO light is emitted either upwards or sideways. (This is known as a Full cut-off (FCO) lighting.)
Larger animals like birds, reptiles, and mammals can simply use gravity to tell them which way is up. Insects, by contrast, are apparently so small that the air itself acts like a viscous medium, and tends to overpower the cues from gravity, much like scuba divers can get confused as to which way is up — unless they follow cues like air bubbles and where the diffused light from the surface comes from.
“The largest flying insects, such as dragonflies and butterflies, can leverage passive stability to help stay upright 30, 31. However, the small size of most insects means they travel with a lower ratio of inertial to viscous forces (Reynolds number) compared with larger fliers32. Consequently, smaller insects, such as flies, cannot glide or use passive stability, yet must still rapidly correct for undesired rotations33. Multiple visual and mechanosensory mechanisms contribute to the measurement and correction of undesired rotations, but most measure rotational rate rather than absolute attitude 26, 28, 32, 34. In environments without artificial light, the brightest portion of the visual environment offers a reliable cue to an insect’s current attitude.”
“Inversion of the insect’s attitude (either through roll or pitch) occurred when the insect flew directly over a light source (Fig. 1 c & Supp. Video 3), resulting in a steep dive to the ground. Once below the light, insects frequently righted themselves, only to climb above the light and invert once more. During these flights, the insects consistently directed their dorsal axis towards the light source, even if this prevented sustained flight and led to a crash.”
The researchers report that certain types of insects did **not** appear to get confused by lights at night: Oleander Hawkmoths (Daphnis nerii) and fruit flies (drosophila).
Because of the wet weather and clouds predicted for Saturday, April 29, 2023 we are canceling the free, public open house we had planned for tomorrow night — itself a postponement because of the clouds on the previous Saturday.
We will try again in the fall.
This is what the GFS forecast is producing for a forecast of average cloud cover for 8 pm EDT Saturday (00:00 Sunday, Universal Time, aka Zulu time) in the mid-Atlantic sector. White means ‘Overcast’.
Come to Bull Run Mountain for a free night under the stars looking at a variety of targets using the telescopes at the Hopewell Observatory on Saturday, April 27, 2024.
You are invited, but will need to RSVP and, in this litigious age, must agree to a waiver of liability for anything that might happen out there in the woods – and they do exist! Plus we don’t have running water — so, we use an outhouse.
But if you take the risk you, for free, can view Jupiter and its moons, comet 12P/Pons–Brooks, and a bunch of bright open clusters like the Pleiades, Beehive and Orion star clusters — and a gaggle of galaxies and double stars.
We have a variety of permanently-mounted and portable telescopes of different designs, some commercial and some made by us, some side-by-side, enabling several people to view the same object in the sky with different magnifications.
The date is Saturday, April 27. We suggest arriving near sundown, which will happen near 8 pm. It will get truly dark about an hour later.
There are no street lights near our observatory, other than some dimly illuminated temporary signs we put along the path, so you will probably want to bring a flashlight of some sort.
If you own a scope or binoculars, feel free to bring them!
Hopewell is about 45 minutes by car from where I-66 intersects the DC beltway. The last two miles of road are dirt and gravel, and you will need to walk about 200 meters/yards from where you park. We do have electricity, and a heated cabin, but since we have no running water, we have an outhouse and hand sanitizer instead.
We are located about 30 miles west of the Beltway on Bull Run Mountain – a ridge that overlooks Haymarket VA from an elevation of 1100 feet, near the intersection of I-66 and US-15. Detailed directions are below.
Assuming good weather, you’ll also get to see the Milky Way itself, although not as well as in years past, because of ever-increasing light pollution.
If you like, you can bring a picnic dinner and a blanket or folding chairs, and/or your own telescope binoculars, if you own one and feel like bringing them. We have outside 120VAC power, if you need it for your telescope drive, but you will need your own extension cord and plug strip. If you want to camp out or otherwise stay until dawn, feel free!
If it gets cold, our Operations Building, about 40 meters north of the Observatory itself, is heated, and we will have the makings for tea, cocoa, and coffee.
Warning: While we do have bottled drinking water and electricity and we do have hand sanitizer, we do not have running water; and, our “toilet” is an outhouse of the composting variety. At this time of year, the bothersome insects haven’t really taken off but feel free to use your favorite bug repellent, (we have some) and check yourself for ticks after you get home.
The road up here is partly paved, and partly gravel or dirt. It’s suitable for any car except those with really low clearance, so leave your fancy sports car (if any) at home. Consider car-pooling, because we don’t have huge parking lots.
Two of our telescope mounts are permanently installed in the observatory under a roll-off roof. One is a high-end Astro-Physics mount with a 14” Schmidt-Cassegrain and a 5” triplet refractor. The other was manufactured about 50 years ago by a firm called Ealing, but the motors and guidance system were recently completely re-done by us with modern electronics using a system called OnStep. We didn’t spend much cash on it, but it took us almost a year to solve a bunch of mysteries of involving integrated circuits, soldering, torque, gearing, currents, voltages, resistors, transistors, stepper drivers, and much else.
We could not have completed this build without a lot of help from Prasad Agrahar, Ken Hunter, the online “OnStep” community, and especially Arlen Raasch. Thanks again!
OnStep is an Arduino-based stepper-motor control system for astronomical telescopes. For this niche application, OnStep uses very inexpensive, off-the-shelf components such as stepper motors and their controller chips — which were developed previously for the very widespread 3-D printing and CNC machining industry.
Getting this project to completion took us nearly a full year of hard work!!! The original, highly accurate Byers gears are still in place, but now we can control the mount from a smart phone!
We also have two alt-az telescopes, both home-made (10” and 14”) that we roll out onto our lawn, and a pair of BIG binoculars on a parallelogram mount.
The drive is about an hour from DC. After parking at a cell-phone tower installation, you will need to hike south about 200 meters/yards to our observatory.
Physically handicapped people, and any telescopes, can be dropped off at the observatory itself, and then the vehicle will need to go back to park near that tower. To look through some of the various telescopes you will need to climb some stairs or ladders, so keep that in mind when making your plans.
Our location is nowhere near the inky dark of the Chilean Atacama or the Rockies, but Hopewell Observatory is mostly surrounded by nature preserves maintained by the Bull Run Mountain Conservancy and other such agencies. Also, our Prince William and Fauquier neighbors and officials have done a fair job of insisting on smart lighting in the new developments around Haymarket and Gainesville, which benefits everybody. So, while there is a bright eastern horizon because of DC and its VA suburbs, we can still see the Milky Way whenever it’s clear and moonless. “Clear Outside” says our site is Bortle 4 when looking to our west and Bortle 6 to our east.
DIRECTIONS TO HOPEWELL OBSERVATORY:
[Note: if you have a GPS navigation app, then you can simply ask it to take you to 3804 Bull Run Mountain Road, The Plains, VA. That will get you very close to step 6, below.]
(1) From the Beltway, take I-66 west about 25 miles to US 15 (Exit 40) at Haymarket. At the light at the end of the ramp, turn left (south) onto US 15.
(2) Go 0.25 mi; at the second light turn right (west) onto VA Rt. 55. There is a Sheetz gas station & convenience store at this intersection, along with a CVS and a McDonald’s. After you turn right, you will pass a Walmart-anchored shopping center on your right that includes a number of fast- and slow-food restaurants. After that you will pass a Home Depot on the right.
(3) After 0.7 mi on Va 55, turn right (north) onto Antioch Rd., Rt. 681, opposite a brand-new housing development. You will pass entrances for Boy Scouts’ Camp Snyder and the Winery at La Grange.
(4) Follow Antioch Rd. to its end (3.2 mi), then turn left (west) onto Waterfall Rd. (Rt. 601), which will become Hopewell Rd after you cross the county line.
(5) After 1.0 mi, bear right (north) onto Bull Run Mountain Rd., Rt. 629. This will be the third road on the right, after Mountain Rd. and Donna Marie Ct. (Do NOT turn onto Mountain Road, and note that some apps show a non-existent road, actually a power line, in between Donna Marie Ct. and Bull Run Mtn. Rd.) Bull Run Mtn Rd starts out paved but then becomes gravel, and rises steadily.
(6) In 0.9 mi, on BRMtn Road, you will see a locked stone gate and metal gate, labeled 3804. That is not us! Instead, note the poorly-paved driveway on the right, with the orange pipe gate swung open and a sign stating that this is an American Tower property. We use their road. Drive through both orange gates, avoiding potholes keeping at least one tire on the high spots. We’ll have some signs up. This is a very sharp right hand turn.
(7) Follow the narrow, poorly-paved road up the ridge to the cell phone tower station.
(8) Park your vehicle in any available spot near that tower or in the grassy area before the wooden sawhorse barrier. Then follow the signs and walk, on foot, the remaining 300 yards along the grassy dirt road, due south, to the observatory. Be sure NOT to block the right-of-way for any vehicles.
(9) If you are dropping off a scope or a handicapped person, move the wooden barrier out of the way temporarily, and drive along the grassy track to the right of the station, into the woods, continuing south, through (or around) a white metal bar gate. (The very few parking places among the trees near our operations cabin, are reserved for Observatory members and handicapped drivers.) If you are dropping off a handicapped person or a telescope, afterwards drive your car back and park near the cell phone tower.
Please watch out for pedestrians, especially children!
In the operations cabin we have a supply of red translucent plastic film and tape and rubber bands so that you can filter out everything but red wavelengths on your flashlight. This will help preserve everybody’s night vision.
The cabin also have holds a visitor sign-in book; a first aid kit; a supply of hot water; the makings of hot cocoa, tea, and instant coffee; hand sanitizer; as well as paper towels, plastic cups and spoons.
The location of the observatory is approximately latitude 38°52’12″N, longitude 77°41’54″W. The drive takes about 45 minutes from the Beltway. A map to the site follows. If you get lost, you can call me on my cell phone at 202 dash 262 dash 4274.
With an Amazon Fire Tablet, on which I placed SkySafari Pro ($15), we can now get the OnStep mount at Hopewell Observatory to go to any target we want, without any wire connection needed at all. The ‘Smart’ Hand Controller is no longer a necessity, which is good, because it’s always been rather a PITA.
The SkySafari Pro interface is really nice and much more user-friendly than any other planetarium software I’ve tried so far. Among other things, you can use your fingers to pan around and zoom into the sky map display, and double tap on a target of interest. Once you’ve located your target on your screen, you can then press ‘GoTo’, and the scope will begin slewing to that target. While it’s doing so, you can watch where the telescope is currently pointing to on the screen’s display, kind of like those airplane icons on maps on some airline flights – only a lot more accurate and zoomable. BTW the connection is via WiFi.
Once the scope thinks it has arrived at the proper location, you can look through the eyepiece (or at a display screen) to see if it is properly centered. If not, then in order to center it, you simply tilt the tablet in the direction you want the scope to go! And changing the speed of such movement is really easy!
I have thanked Arlen for showing me this on his cell phone. I myself could never get it to work properly with my iphone, but after some time downloading the proper software onto the tablet and making the proper wifi connections with the proper IP address and port number, in a nice warm location here in town with at least a halfway decent WiFi connection, with a spare OnStep setup on the bench in front of me, then it was easy.
I demonstrate this with the following clumsy video.
BTW, SkySafari Pro works on Android and other tablets, on MacOS, Windows, and supposedly even on iPhones. You do need to pay for the Pro version, because the free version does not have telescope control capabilities.
So, for very little money, but a whole lot of work, we have 21st-century Wi-Fi control over a very fine telescope mount!
Hopewell Observatory is once again holding a free, public, Autumn observing session, and you are invited.
You and your friends and family can get good looks at the planets Saturn and Jupiter, as well as a bunch of open and globular star clusters. And there will be a gaggle of galaxies and double stars to look at as well.
We have a variety of permanently-mounted and portable telescopes of different designs, some commercial and some made by us, some side-by-side. Two or three people can view the same object in the sky, through different optics, with different magnifications, all at the same time! The differences can be quite amazing…
You will be capturing those photons with your own eyes, in real time, as they come to you from however far away, instead of looking at someone’s super-processed, super-long-exposure, false-color, astro-photograph (as beautiful as that image may be).
We suggest arriving near sundown, which will occur around 6 pm on 11/4/2023. It will get truly dark about 7:30 pm. The waning, last-quarter Moon won’t rise above the trees until roughly midnight. While beautiful, the Moon’s light can be so bright at Hopewell that it casts very obvious shadows, and this of course tends to make distant nebulae and our own Milky Way harder to see., so we will have many hours of Moon-free observing if the weather holds up.
If it is hopelessly cloudy and/or rainy and/or snowing, we will cancel and reschedule.
There are no street lights near our observatory, other than some dimly illuminated temporary signs we hang along the path, so you will probably want to bring a flashlight of some sort. Your cell phone probably has a decent one, but it’s better if you can find a way to cover the white light with a small piece of red plastic tape– it will save your night vision.
If you own a scope or binoculars, feel free to bring them, and you can set it/them up on our lawn.
Hopewell is about 30 miles (~45 minutes) by car from where I-66 intersects the DC beltway, but rush hour gridlock can double that time, easily. The observatory is located atop Bull Run Mountain – a ridge that overlooks Haymarket VA from an elevation of 1100 feet, near the intersection of I-66 and US-15. The last two miles of road are dirt and gravel, and you will need to walk about 250 meters/yards from where you park. We do have electricity, and a heated cabin, but since we have no running water, we have an outhouse and hand sanitizer instead.
Detailed directions are below.
Assuming good weather, you’ll also get to see the Milky Way itself, although not as well as in years past, because of ever-increasing light pollution.
If you like, you can bring a picnic dinner and a blanket or folding chairs, and/or your own telescope binoculars, if you own one and feel like bringing them. We have outside 120VAC power, if you need it for your telescope drive, but you will need your own extension cord and plug strip. If you want to camp out or otherwise stay until dawn, feel free!
If it gets cold, our Operations Building, about 40 meters north of the Observatory itself, is heated, and we will have the makings for tea, cocoa, and coffee.
Cautions
Warning: While we do have bottled drinking water and electricity and we do have hand sanitizer, we do not have running water; and, our “toilet” is an outhouse of the composting variety. At this time of year, it’s often too cold for many of the nastier insects, feel free to use your favorite bug repellent, (we have some), tuck your pants legs into your socks, and check yourself for ticks after you get home.
The road up here is partly paved, and partly gravel or dirt. It’s suitable for any car except those with really low clearance, so leave your fancy sports car (if any) at home. Consider car-pooling, because we don’t have huge parking lots.
Our Telescopes
Two of our telescope mounts are permanently installed in the observatory under a roll-off roof. One is a high-end Astro-Physics mount with a 14” Schmidt-Cassegrain telescope made by Celestron and a 5” triplet refractor by Explore Scientific. The other mount was manufactured about 50 years ago by a firm called Ealing, but the motors and guidance system were recently completely re-done by us with modern electronics using a system called OnStep, after the old gear-and-clutch system died. We didn’t spend much cash on the conversion, but it took us almost a year to solve a bunch of mysteries of involving integrated circuits, soldering, torque, gearing, currents, voltages, resistors, transistors, stepper drivers, and much else.
We could not have completed this build without a lot of help from Prasad Agrahar, Ken Hunter, the online “OnStep” community, and especially Arlen Raasch. Thanks again! (OnStep is an Arduino-based stepper-motor control system for astronomical telescopes that uses very inexpensive, off-the-shelf components such as stepper motors and their controller chips that were developed previously for the very widespread 3-D printing and CNC machining industry. The software was written by Howard Dutton. Thanks, Howard!)
The original, highly accurate Byers gears are still in place, but now it’s not just a Push-To-and-Track scope, but a true Go-To mount with very low periodic error that we can run from a smart phone! On this incredibly rugged scope mount we have two long-focal-length 6″ refractors by Jaegers and D&G, a home-made short-focal-length 5″ refractor, and a 10″ Meade SCT.
We also have two alt-az (Dob-mounted) telescopes, 10″ and 14″, both home-made, that we roll out onto our lawn, and a pair of BIG binoculars on a parallelogram mount.
Both the observatory building and the operations cabin were completely built by the hands of the original founders, starting in the early 1970s. This included felling the trees, bulldozing the clearing, planning and pouring the foundations, laying the concrete blocks, welding the observatory’s roll-off roof, and repurposing a bomb hoist to open and close that roof. Many of the founders (Bob McCracken, Bob Bolster, Jerry Schnall in particular) have passed away, but we current members continue to make improvements both small and large. In the Operations Cabin, you can see some wide-field, film astrophotos that Bolster made, and the Wright-Newtonian scope that he built and used to make those images.
Access
After parking at a cell-phone tower installation, you will need to hike south about 250 meters/yards to our observatory. Physically handicapped people, and any telescopes, can be dropped off at the observatory itself, and then the vehicle will need to go back to park near that tower. To look through some of the various telescopes you will need to climb some stairs or ladders, so keep that in mind when making your plans.
Our location is nowhere near the inky dark of the Chilean Atacama or the Rockies, but Hopewell Observatory is partly surrounded by nature preserves maintained by the Bull Run Mountain Conservancy and other such agencies, and our neighbors on both sides of the ridge have never been a problem. Unfortunately, the lights in Gainesville and Haymarket seem to get brighter every year. “Clear Outside” says our site is Bortle 4 when looking to our west (towards the mountains) and Bortle 6 to our east (back into the suburban sprawl).
DIRECTIONS TO HOPEWELL OBSERVATORY:
[Note: if you have a GPS navigation app, then you can simply ask it to take you to 3804 Bull Run Mountain Road, The Plains, VA. That will get you very close to step 6, below.]
Otherwise:
(1) From the Beltway, take I-66 west about 25 miles to US 15 (Exit 40) at Haymarket. At the light at the end of the ramp, turn left (south) onto US 15.
(2) Go 0.25 mi; at the second light turn right (west) onto VA Rt. 55. There is a Sheetz gas station & convenience store at this intersection, along with a CVS and a McDonald’s. After you turn right, you will pass a Walmart-anchored shopping center on your right that includes a number of fast- and slow-food restaurants. After that you will pass a Home Depot on the right.
(3) After 0.7 mi on Va 55, turn right (north) onto Antioch Rd., Rt. 681, opposite a brand-new housing development called Carter’s Mill.
(4) On Antioch Rd. you will pass entrances for Boy Scouts’ Camp Snyder and the Winery at La Grange. Follow Antioch Road to its end (3.2 mi), then turn left (west) onto Waterfall Rd. (Rt. 601), which will become Hopewell Rd after you cross the county line.
(5) After 1.0 mi, bear right (north) onto Bull Run Mountain Rd., Rt. 629. This will be the third road on the right, after Mountain Rd. and Donna Marie Ct. (Do NOT turn onto Mountain Road. Also note that some apps show a non-existent road, actually a power line, in between Donna Marie Ct. and Bull Run Mtn. Rd.) Bull Run Mtn Rd starts out paved but then becomes gravel, and rises steadily.
(6) At 0.9 mile on Bull Run Mountain Road, you will see a locked stone gate and metal gate, on your left, labeled 3804. That is not us! Instead, note the poorly-paved driveway on the right, with the orange pipe gate swung open and a sign stating that this is an American Tower property. We will also put up a temporary, lighted sign to Hopewell Observatory. (We have long-standing permission to use the cell tower’s access road). This is a very sharp right hand turn.
(7) Follow the narrow, poorly-paved road up the ridge to a fenced-off cell phone tower station. Drive through both orange gates. Try to avoid potholes. In places where there is a high ridge between the tire tracks, I suggest you NOT try to straddle the ridge. Instead, straddle the low spot, and drive with one set of tires riding on the high central ridge.
(8) Park your vehicle in any available spot near that cell phone tower or in the grassy area before the wooden sawhorse barrier. Then follow the signs and walk, on foot, the remaining 250 yards along the grassy dirt road, due south, to the observatory. Be sure NOT to park in such a way that your vehicle will block the right-of-way for any other vehicle.
(9) If you are dropping off a scope or a handicapped person, move the wooden barrier out of the way temporarily, and drive along the grassy track into the woods, continuing south, bypassing a white metal bar gate. (The very few parking places among the trees near our operations cabin, are reserved for Observatory members and handicapped drivers.) If you are dropping off a handicapped person or a telescope, afterwards drive your car back and park near the cell phone tower, and put the barrier back into place. Thanks.
Please watch out for pedestrians, especially children!
In the operations cabin we have a supply of red translucent plastic film and tape and rubber bands so that you can filter out everything but red wavelengths on your flashlight. This will help preserve everybody’s night vision.
The cabin also holds a visitor sign-in book; a first aid kit; a supply of hot water; the makings of hot cocoa, tea, and instant coffee; hand sanitizer; as well as paper towels, plastic cups and spoons.
The location of the observatory is approximately latitude 38°52’12″N, longitude 77°41’54″W.
A map to the site follows.
If you get lost, you can call me (Guy) on my cell phone at 202 dash 262 dash 4274 or email me at gfbrandenburg at gmail dot com.
A decade or so ago, I bought a brand-new Personal Solar Telescope from Hands On Optics. It was great! Not only could you see sunspots safely, but you could also make out prominences around the circumference of the sun, and if sky conditions were OK, you could make out plages, striations, and all sorts of other features on the Sun’s surface. If you were patient, you could tune the filters so that with the Doppler effect and the fact that many of the filaments and prominences are moving very quickly, you could make them appear and disappear as you changed the H-alpha frequency ever so slightly to one end of the spectrum to the other.
However, as the years went on, the Sun’s image got harder and harder to see. Finally I couldn’t see anything at all. And the Sun got quiet, so my PST just sat in its case, unused, for over a year. I was hoping it wasn’t my eyes!
I later found some information at Starry Nights on fixing the problem: one of the several filters (a ‘blocking’ or ‘ITF’ filter) not far in front of the eyepiece tends to get oxidized, and hence, opaque. I ordered a replacement from Meier at about $80, but was frankly rather apprehensive about figuring out how to do the actual deed. (Unfortunately they are now out of stock: https://maierphotonics.com/656bandpassfilter-1.aspx )
I finally found some threads on Starry Nights that explained more clearly what one was supposed to do ( https://www.cloudynights.com/topic/530890-newbie-trouble-with-coronado-pst/page-4 ) and with a pair of taped-up channel lock pliers and an old 3/4″ chisel that I ground down so that it would turn the threads on the retaining ring, I was able to remove the old filter and put in the new one. Here is a photo of the old filter (to the right, yellowish – blue) and the new one, which is so reflective you can see my red-and-blue cell phone with a fuzzy shiny Apple logo in the middle.
This afternoon, since for a change it wasn’t raining, I got to take it out and use it.
Two days ago, Joe Spencer had first light with the 6″ f/8 Dobsonian he built in the DC-area amateur telescope workshop. He worked hard on this project over more than a year, including grinding, polishing and figuring his mirror, and it seems to work very well.
Fascinating article about how the origins of the telescope are not quite so clear after all. (Pun intended!)
(EDIT: Roger wanted me to emphasize that he doesn’t think he has the final word on this history. Later, I’m going to try to make his graphics more visible, too. Looks like WordPress has changed how you add photos again!)
More on the Origins of the “Telescope.” LONG! From: Roger Ceragioli Date: Mon, 25 Jul 2022 21:30:48 EDT
Greetings again. The Galilean telescope is a peculiar beast optically. Part of the reason it was the first form of telescope was that it gave an erect image, using only two lenses. But because there is a particular third optical element (the human eye) involved, we need to consider the total sytem (convex objective, concave eyepiece, human eye) to understand what may have happened in 1608 as well as in the decades prior.
And so in this email, I want to review some aspects of this “bionic” system. Part of what has, in my opinion, afflicted the debate over the origins of the telescope is lack of attention to the nature of this total bionic system.
Now, it is, of course, hardly conceivable that if the first eyeglasses came into use around 1295 AD, using convex lenses to treat long-sightedness (presbyopia), and concave lenses came into use around 1450 to treat myopia, for 150 years no one bothered to experiment with placing different types of lenses together to find out what would happen. As I mentioned previously, we read in Fracastoro’s book, Homocentrica (a book on planetary theory) about the marvelous effects of putting one convex lens on top of another. Apparently this means directly on top of one another, and not at a distance. Still it shows that people were putting lenses together in the 16th c.
Giovanni Battista della Porta in 1589 mentions combining a convex with a concave (ie. the essence of an early telescope), and that if you know how to do it right, both nearsighted and farsighted people can see clearly. So I would propose, not that he “invented” the telescope, but that he took a step forward and invented a kind of variable focus low power two-lens eyeglass system. It could have been constructed as follows:
Pay no attention to the title of the slide, when it says “Galilean-Type Refractor.” Instead, notice the magnification: 1.25x (a VERY low power), and how the device is constructed. You have on left a 4-diopter (10-inch focus) convex lens, good for old people to do close-up work. And on left a 5-diopter concave, good for strongly myopic people. You do the viewing from right to left through the eyepiece, placing your eye’s pupil as shown.
Now, part of the reason why eyeglass lenses can be optically quite terrible and work just fine, is that in daylight the eye’s pupil closes down to about 2 or 3 mm in diameter. In the dark at night it opens to 5, 6, 7 or even 8 mm, depending on age, to let in more light. The above diagram assumes 2 mm. The blue, green, and red lines passing from left to right represent NOT colors of light, but different ray bundles coming from differing directions out of the graphic on left. I.e. different field positions. Blue is on-axis, green is 2 degrees off-axis, and red is 4 degrees off-axis. The bundles all converge on the eye’s biological pupil and are focused onto the retina.
What’s important here is that in this sytem, the eye’s pupil becomes the delimiting factor in determining how much light enters the eye from each object point. By one definition of magnification,
Magnification = Diameter of Entering Bundles/Diameter of Exiting Bundles in a telescope.
By measuring the bundle diameters we derive the magnification. So, for example, if you have a 200 mm telescope mirror, then the entering bundle is 200 mm in diameter. And if the exiting bundle is only 1 mm in diameter after it comes out of your eyepiece, then we can say that your telescope’s magnification is 200x. In the diagram above, if the magnification is 1.25x and the bundles of rays passing out of this “telescope” and into your eye are each 2 mm in diameter, then by the equation the bundles entering the objective must by 2 x 1.25 = 2.5 mm.
So in the above system, the objective lens doesn’t need to be any better than a common eyeglass lens for the viewer to find the (barely magnified) scene perfectly sharp. It would, therefore, have been easy for Della Porta to make the above device and have it work just as he says. In 1609, when asked, he sent the following rough sketch:
Crude, but it sure looks like a telescope. The “c” tube “trombones” in and out to focus. By this stage 20 years later, Porta was sneering at the device and calling it “crap.” Crap in the sense that it was amusing, but had little effect on “making things nearer.” With my 1.25x version above, the length of the device would be about 100 mm.
The image quality (assuming decent glass) would be perfect all over the field:
Here the colored dots do represent colors of light. The black circles represent the size of the Airy disks. Since the dots for the three sample colors (486, 546, and 656 nm representing the visual spectrum) all fall well within the black circles, we can surmise that a viewer using the system will see everything sharp. The device is only of interest in showing the basic geometry of a Galilean-style refractor, and in that it can focus to varying distances, from infinity to about 2 meters. Anyone can use it, since like a telescope no matter what the state of your eyes (nearsighted or farsighted) you can refocus. But the magnification is hardly noticeable.
Now, if you try to increase the magnification, then the difficulties arise. Let’s say we replace the eyepiece with one that magnifies to 6x:
Here I have stretched the drawing vertically by 5x versus reality to make the paths of the rays for different field positions clearer. The first thing that’s happened is that because of the 6x magnification, now the entering bundles are 2 x 6 = 12 mm in diameter! For each field position, the entering rays cover much more of the objective lenses surface now, just as in a real telescope. But, indeed, this IS now a real telescope!
The result is that any appreciable defects in the objective lens WILL degrade the image. If the glass is of low quality in transmission, or if the lens surfaces are not very spherical, they will blur the images, just as the objective in your childhood “Trashco” telescope used to do. It is these problems that Rolf Willach rightly pointed to in his book, The Long Route to the Invention of the Telescope. But there are still more that he did not discuss.
Because in this type of optical system the “pupil” (ie. where all the ray bundles intersect) as at the eye, and there is a lot of refraction far away at the objective lens, you will inevitably get “lateral color” in the images:
Here we have a spot diagram for the 6x system. On-axis we’re ok. But off-axis at 0.125 and 0.25 degree (the true field of view is now much smaller than before), the red, green and blue bundles are decentered from one another. That means that stars will be seen stretched and smeared, ie. very unsharp. Also, on-axis we now see longitudinal chromatic aberration, although not bad. This system won’t work well as a telescope, because most of the field of view will be smeared. It is this effect (as well as any smearing from surface figure error or glass inhomogeneity) that Lipperhey needed to correct.
He did so, it appears to me, probably by imposing a diaphragm on his objective lens. We don’t know this for certain, but it is very likely, as Willach first suggested. Now, the important part to understand is that if the objective diaphragm makes the bundles of rays exiting the eyepiece smaller than the eye’s own pupil, then what Lipperhey really did was to transfer the pupil of the bionic system from the human eye to the objective lens (where it should be for a telescope). This transference instantly sharpens images in the outer field:
The system layout and raypaths now look like so:
The entering bundles all intersect at the objective and diverge where the eye pupil goes. So we get much sharper images, with much reduced chromatic effects, but at the cost of a narrow, very narrow field of view. Galilean telescopes are infamous for their narrow fields of view. You have to “scrunch” your eyeball up against the eyepiece glass and move it side to side to see as much field as possible. This is inevitable given the optics. And it all gets worse in general as the magnification increases.
If you want to reach even 20x or 30x, it’s necessary to make the telescope longer and longer, to mitigate the chromatic effects. My 6x system above is only about 400 mm long, similar to terrestrial telescopes seen in early images. But for an astronomical telescope, magnifying 20 or 30x, you might need triple or quadruple of that. With triple or quadruple of the objective size. That is 24 or 32 mm. Galileo’s famous 1610 telescope with which he found Jupiter’s largerst moons, seems to have had an aperture of 38 mm. It required a mounting (as Galileo advised) to hold it steady.
To conclude, we have a number of other people aside from Lipperhey who claimed to have used a device similar to his before him. Certainly, Della Porta did in 1589. And perhaps also, Jacob Metius of Alkmaar, as well as an unidentified “young man” in Holland. It may be that Raffaello Gualterotti had something like this in Italy, and Joan Roget in Spain, as well as others. Likely, if these devices really existed, they were of very low power. Metius admitted that his didn’t work too well. But the point here is that I think these claims should not all be dismissed as fraud or sour grapes.
We see above why such devices could easily have existed and probably did before 1608. And yet we also see why they would in general have failed, if their authors tried to increase the magnification. It required that the system pupil be transferred from the eye to the objective lens, before you could get a functioning telescope of notable magnification. The reason is not only what Rolf Willach has rightly pointed too (glass quality and surface/wavefront figure), but also because of the underlying optics of chromatic aberration.
The same thing happened in the 1630s, when the Keplerian telescope was adapted for terrestrial viewing. Its output image had to be inverted and reversed. Kepler himself had suggested a method already in 1611. But in practice this led to terrible results, due to lateral color error, smearing off-axis stars. Let us remember that chromatic aberration as such was not recognized until Isaac Newton in the 1670s. Before then people had no idea that light is not intrinsically white. They thought colors were some kind of modification of white light. But what kind of modification, and how it all worked utterly flummoxed them. So there was no meand to intentionally correcting color error. No theory or solutions could exist until much later.
And so, just as I suspect Lipperhey may have hit upon a solution to his problem through the use of a diaphragm, possibly under the influence of a false (then widespread) theory of how the eye works, so too when Anton Maria Schyrleus de Rheita, and later Giuseppe Campani found the first effective 3-lens terrestrial eyepieces, which completely corrected the lateral chromatic aberration plaguing Kepler’s 2-lens eyepiece, they couldn’t know the true reasons why their systems worked. Yet work they did.
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