Greek Physics: Calculating the distance to the Sun and Moon
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- Опубліковано 23 жов 2018
- This video looks at the method of Aristarchus for determining the distance to the Sun and Moon.
This simplified calculations in this video follow the presentation given in 'To Explain the World' by Steven Weinberg, which provides an excellent commentary on the discovery of modern science.
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The fact that they got it right within 2 degrees using nothing but sticks is just a sign of how incredibly intelligent these great humans were.
Aren't humans always good when they aren't limited?
and today people think the Earth is flat ..
It's simply amazing..And the fantastic Cropcircles that these Greeks could draw with those sticks, it's just remarkable.And your not even discussing how easy they found Water with those twisted sticks. These Greeks...Their Too Much...
a bunch of bored men with olive oil can accomplish great feats
@@swedneck you mean Nomads.
While children are out of school this is a great recreative way to learn sciences being geometry, mathematics and astronomy. The most important are the expressions, then as measurements become more and more accurate you can plug better numbers into them.
I completely agree, I think it is a great way for students to learn about the basics of trigonometry and how simple ideas in Maths can be related to the natural world
nice speaking, greetings from brazil, in quarentene too
@@lecinquiemeroimage I was questioning the measure of this angle θ : at that time who would be able to correct the atmospheric optical deviations ?
@@AlexandreLollini nobody, that's why they were off in 300 B.C. - but as measurements became more and more precise, the results became more and more accurate.
also note: strictly speaking, the geometry shown here uses some simplifications (e.g. treating arc of large circles as straight lines, assuming that the moon exactly covers the sun during an eclipse, etc...). But the errors are relatively small in the grand scheme of things.
HenryDavidT They maybe way off but the fact they got such a reasonable approximation is glorious. They used mathematics to fill the gaps a lack of technology created.
As an astronomer who has calculated Earth's circumference using Eratosthenes' method (by sighting the North Star from two locations), your video gave me great delight. It is indeed wonderful that such a calculation could be made, at least in principle, using only well timed observations of the Earth Sun and Moon. Here are two clarifications:
Part 1: The 90 degree triangle has two angles that are 90 degrees! (Almost) This measurement was the hardest part because you need to point who sticks at the Sun and first quarter moon and measure the angle between them AND you need to correctly guess the exact moment when the moon is at 1st quarter (creating one of the 90 degree angles). You better hope this happens during the daytime also. This is probably why an accurate distance to the Sun was not computed until the Transit of Venus in 1769.
Part 2: The premise of this part is only valid if the angular size of the Moon and Sun are the same. Evidence that this is approximately true comes from the fact that some solar eclipses are NOT total but annular, while others are totally. This means that the Moon's distance is very close to the right distance to have the same angular size of the Sun. The variation is caused by the fact that both Moon's orbit, and Earth's are ellipses.
Your viewers may also be interested this video which could be subtitled "If Eratosthenes had a bike":
ua-cam.com/video/YaPa4esJJx4/v-deo.html
@Allan 112358 He need only watch a series of predictable lunar eclipses. The umbra (shadow) of Earth as it covers the moon always shows a similarcurve and as such indicates a constant radius. This does not require a total Lunar eclipse, but merely the presence of a good cross-ection of the umbra to see clear evidence of its circular shape. The true deviation from a perfect sphere is so small as to be invisible to the human eye by that method; amounting to less than one part in 800.
@Allan 112358 Oh Please. A flat disk may cast a round shadow only if the axis of its its plane is pointing near the source of illumination. So, Firstl, What makes you think a flat Earth must be a disk? If the planet is "flat" then a curved edge becomes neither required or even likely. Second, if the disk were facing the the Sun all the time, as were requited for the shadow to be curvelinear, then there would not be night and day cycles on Earth: Since we know there are night and Day cycles, and that Lunar Eclipses can happen at Any time of Day or night....You see the problem there?
Yeah that angle seemed pretty arbitrarily chosen...
I'm much wondering how they managed to make any guess at all as it boils down to distinguishing between perfect 90° half moon and 89.85° almost half moon, rendering distance to the Sun apparent infinity.
@@photonjones5908 -- I guess I missed the dialog you had with @Allen 112358, whom I presume to be a flat earther, but it looks like he/she/it deleted his/her/its posts.
Beautiful example of the power of mathematics and human ingenuity. Really clear explanation of how these results were obtained.
Glad you enjoyed it!
@@PhysicsExplainedVideosHey this was an excellent explanation and a fantastic video, but I have a question, how did Aristarchus figure out the angle of Φ was approximately 87 degrees. I’m honestly just impressed with how he was even capable of doing such an achievement using only sticks. But anyway it’s still an amazing video that helped me to study Geometry in a fun and creative way
This was my favorite proof from the history of astronomy book while I was studying. Very well explained and presented!
And to think that there are people who 2000+ years after these calculations were performed still think the world is flat.... makes me sad.
Yes it’s amazing. It’s so simple but the FE ers are impossible to convince.
Honestly, I think FE know Earth is NOT flat, they pretend and fight themselves to believe it is, so they can make money with youtube videos.
@@ariesmars29 There are clearly flat earthers who belong to the category you describe.
But that would not work, unless there had also been a large number of flat earthers who genuinely believe that the earth is flat.
@walt7500 The reason why we should care about what flat earthers believe, is that a large number of them also believe in other conspiracies, such as anti-vaxxing. And some of these conspiracy theories are downright dangerous.
The modern flat earth delusion was started in england in 1838 when Samuel rowbotham used a telescope to see if a boat sailing down 10km of the old Bedford river would disappear under the horizon. Apparently he didn't see it disappear and so started spreading the flat earth delusion.
The first flat earth society was created in england 1956 by shenton.
The key takeaway here is that it started in western countries and it was spread by westerners all around the world. Other backward ideas are similar, catholicism, protestantism, capitalism, marxism, white supremacy, etc.
There's something very wrong with westerners.
I could never do this calculation through by myself but it was certainly a thrill to see it develop as you walked us through. Thank you and I thank the ancient Greeks too.
It's even more incredible that this was don when people were still using sticks and compasses to calculate.
This also seems like a fun way to spend our quarantine time.
Bjarni Valur they also had the abacus, and geometry.
These guys were idiots. The sun is less than 3000 miles away. same goes with the moon. And they are the same size.
The Two Gardens - they CAN’T be the same distance. Because one goes BEHIND the other, one must be further away.
Because one is further away, but LOOKS to be the same size, it CAN’T be the same size, it HAS to be larger, or, being further away, it would appear to be smaller.
Even children understand this much. The Greeks realized this and so asked the question... can we determine how much further away the sun is since we absolutely know it’s further than the moon? And with Geometry, they could.
I’ll believe you can outthink the Greeks when you show your calculations.
@@thetwogardens6048 M8 ... are you being serious or are you just takin the piss?
@Bjarni Valur
Try the two setset or sunrise method to determine the Earth’s circumference. It was written up in the American Physics teacher twenty years ago .
I adore this problem from ancient astronomy. I have repeatedly made my own observations of the various pieces of this puzzle, especially several lunar eclipses and sextant measurements of the Sun, Half Moon, and Earth angles (correcting for parallax of course). There is a thorough presentation of Aristarchus's geometrical solution in James Evan's book, The History and Practice of Ancient Astronomy, although the inclusion of Eratosthenes' Earth diameter measurements is not included since they were made after Aristarchus.
What I find interesting is that no one after Aristarchus completed the problem using the value from Eratosthenes' work. The original work of Aristarchus has not survived; but, has been quoted at length by Archimedes' work The Sand Reckoner. Yet, even he did not make the obvious connection. Aristarchus' model was vilified by his work since it violates the dominant Aristotlean paradigm until the 16th century. Nevertheless, it is fascinating that the obvious completion of the problem of Aristarchus was never made, or accounts of such a completion has not survived.
This is a wonderful presentation of the right path that was not taken until recently. Kudos to you of the highest order. It clearly shows that the modern conception of the solar system could have been known in ancient times. Certainly, this is a warning to those who demand we follow the "received wisdom" of our "betters."
Thank you for making this effort.
Very clearly illustrated and articulated a wonderful presentation, well done!
Thanks for the kind words of feedback
I would like to add here that the Greeks did geometry, they did not think they did Physics in the way understood by us today. Still, their discoveries were remarkable, especially those of Archimedes who began to invent calculus using infinitesimals. In my view, dogmatic adherence to a platonic (absolute) world view impeded them, even the great Archimedes, to preempt the advances of the 7th century onwards. Thus, the Ptolemaic system prevailed, introducing ever more unnecessary concepts (epicycles) which provided mathematical accuracy but were way off physical reality. In the 5th century AD another Greek physicist, Iοannis Philoponos (roughly translated as John the Diligent, because he was a hard worker) who refuted Aristotle's physical concepts and was the first to introduce the law of free fall. I am not aware that Galileo knew about his work but he certainly is not the first to propose it. This also refutes prejudice against the Greeks that they did not do experiments. Generally, the idea that Greek science lost its vigor is a misconception and is based on later hostile opinions in medieval times. Worse, Aristotle was distorted and blamed for lack of scientific progress. I am certain that if Aristotle lived he would have been in agreement with those who did not perceive science as dogma. His contribution to Biology (besides the term itself) is monumental even by today's standards.
Refs: Richard Sorabji (Hrsg.): Philoponus and the Rejection of Aristotelian Science. Cornell University Press, Ithaca (New York) 1987
and, Christian Wildberg: John Philoponus' Criticism of Aristotle's Theory of Aether. De Gruyter, Berlin 1988
This is an interesting read. I am not a great fan of Aristotle, but I might look into those books you mention. My view has always been that the ancient Greeks had some truly amazing thinkers that were hamstrung by preconceived notions about the world, and with an unfortunate bias towards rationalism over empiricism. By cherry picking from their theories, you can make them seem wise and prescient beyond belief, but in truth, there are so many random ideas from the period that one can find support for pretty much anything. Like awarding the idea of the atom to Democritus and Leucippus when their metaphysical nonsense has no bearing (or influence) whatsoever of the modern understanding of matter. When it comes to actual understanding of the physical world, the ancient Greeks were completely scattershot. But one should also not underestimate the insanely impressive things some of them came up with. Aristotle was certainly a great (albeit often wildly off the mark) thinker, and if the sheer authority of his teachings (or misappropriations thereof) were perhaps making it hard for original ideas to compete for a very long time, that blame should surely not be laid at his feet. Nothing has ever been more detrimental to the progress of human knowledge than the catholic church. Without their intellectual oppression, we would have been able to build on the Greek knowledge rather than start from scratch a millennium later.
@@egodreas Dear Andreas, in fact, the development of modern science took place within the Universities in Europe under the protection of the Catholic Church. Strange as it may sound, it was Dominican and Franciscan monks, who first progressed towards a separation of faith and logic, in their quest to use Aristotle's iron-clad "faith" in the power of reason to understand nature. Even POpes foresaw the power of Aristotle's method in strengthening religious faith. It is a different matter if that lead to the development of University faculties that gave rise to science as we know it. That is why several centuries after the Greeks in Several figures are prominent, Siger of Brabant, the first to formally the age of scientific reason, followed by Dans Scotus, Roger Bacon, and many others. Siger of Brabant was murdered under obscure circumstances but others survived. The Hellenistic period was a time of great philosophical activity, but unfortunately, most of the primary sources have been lost. Also, there were many Greeks who nowadays are generally ignored in the West such as Ioannis Philoponos (Alexandria, 490-570 AD), a major figure in my view (not because he was Greek, mind you, but because he anticipated Galileo and Newton by more than 1000 years) Michael Psellos (Constantinople, 1018-1078 AD) and Georgios Plethon (Constantinople, 1355-1452 AD). The Greeks up until the 18th century continued the tradition of the antithesis between Platonism and Aristotelianism because they had settled since ancient times and considered these issues as settled in regards to their ontology. Even as late as the 1600s they were writing with the Attic dialect ignoring the progress of the Greek language to its modern form, a direct descendant of ancient Greek.
@@nicka.papanikolaou9475 no, it didn't. The Catholic Church was in power since the 5th century onwards, and the development of modern science was not due to Christianity. Also, when you say “unfortunately, most of the primary sources have been lost” that's an understatement: the _Christians_ decided that they were not interested in the largest part of Greek and Roman science
@@Panairjdde I have never said that it was DUE to Christianity. But fact is, modern resurfaced within the Universities populated by Christian Monks. Scholasticism gave rise to science. Also, the previous speculations of Arab scholars attempted to build on the Greeks. Some contributions were also made by the Indians but we know little.
@@nicka.papanikolaou9475 if the religion was not important, then why underline the fact that they were Christians? Besides, they were not monks, universities had some monks and a lot of laity
From one math teacher to another - you've made these calculations crystal clear; fascinating to see the simple calculations develop to final conclusions! I will definitely share these with my geometry classes. Well done, sir!
It's nice to be able to view these older videos and be able to see the raw talent of simple presentation which is the foundation that the newer prettier videos are built upon.
Eratosthenes would not have missed the problem at 5:44 - which assumes the triangles are similar only because that's the way it's drawn. There is nothing physical that requires the shadow to come to a point at the same distance.
I wonder if the video is not quite right there; a total solar eclipse - in those instances when the Moon more or less exactly covers the Sun - actually gives you nearly that diagram but with the sharp end of the triangles at the surface of the Earth (where the observer’s eye is), not its centre. So the relationship is actually (Rm/(dm+RE)) = (Rs/(ds+RE)) where RE is the radius of the Earth. Given the relatively small value of RE though it probably doesn’t change the end results too much. Fantastic video by the way.
This is very well done. The sorts of approximations and use of properties of triangles and circles underpin many derivations of celestial mechanics properties. This presentation is important in introducing that type of thinking. Newton's Principia is, of course, all geometrical and hard to follow because we are so used to analytical techniques. The visualization in this presentation is the key to its usefulness. But you really do have to give it Aristarchus, Eratosthenes, Archimedes etc. Immensely powerful and practical intellects. I always feel two inches high when I look at their achievements.
In my top 10 on YT. I've recorded and watched this every night for so long. Chapeau.
It is worth noting that an accurate determination of the Earth-Sun distance was absolutely necessary in determining the scale of the Solar system, based on the orbital periods relationship to the distances of the known planets; and eventually led to an accurate and verifiable determination of the speed of light (c) itself, through accurately measuring deviations in the timing of the disappearances and reappearances of the Galilean Moons as they passed behind Jupiter as seen from Earth, during the course of Earth's trip around the Sun. We take it for granted now, but all of this began in the human brain -with the help of two sticks.
Estimation of the theta is key for finding d_s and d_m relationship, any tiny error there will be amplified since it's so close to 90 degrees. I wish you explained where that value comes from.
@Fatih Karakurt
The modern value of the angular measurement from the Moon to the Sun is 89.85 degrees . In the time of the ancients I believe their value was 87 degrees . With their technology available still a good measure .
See how in the 1700s Cook’s voyage one task was to measure the transit of Venus across the Sun as it was also being observed in Europe using the Earth as a baseline .
I'm also struggling to understand how they made that measurement in greek times
@Xavoop
Any common sighting tool will do such as cross staff or a sextant . If I recall correctly the babylonians used stiff clay tablets and place marking in it . All one needs to do is sight on the terminus of a quarter moon phase and the Sun( lower limb ) at the same time .
Ancient Greek tools
kotsanas.com/gb/cat.php?category=13
listverse.com/2015/09/13/10-incredible-astronomical-instruments-that-existed-before-galileo/
How far would a pinhole camera, dark room, an accurate ruler and many solar observations at around midday get you these days?
@jq4t49f3
Well NASA a deep long focal length imager of Sun to track Sun spots and CMEs too !
A pin hold camera follows the
Same equation as a lens .
1/f=1/do+1/di
Where : f is focus
do is distance to the object
di is distance to the image .
en.m.wikipedia.org/wiki/Sunspot_Solar_Observatory
That was so so so so beautiful. I love seeing how mathematics is such a wonder to behold. I’d share my joy to my friends and family but they’d all laugh at me!!! 😄
Clear explanation of the use of geometry to find these distances. Really tied them all together into a neat package.
Thanks for this video! It was lovely to see parts of my 1975 Scottish Certificate of Sixth Year Studies school physics project, so clearly explained. In the end I got results that were a lot worse than the ancient Greeks did, despite the help of clocks and telescopes, and accurate star charts. I couldn't even explain why my figures and error bound estimates, didn't match the currently accepted distances. I knew about spherical trigonometry, but I didn't have the maths in those days, so my approximation and simplification errors grossly overwhelmed my measurement errors. But I still got a prize for experiment concept, and for trying. Naked Harry (Aristarchus) you rock!
You explain this with such an easy thought process to arrive at the conclusions. Thank you for sharing
My pleasure, thanks for the comment
There are currently 110 flat-earthers somewhere around...
Now 132, as of 26th may 2021
Like a track? Race Tracks, and tables are round, not oblate sparical.
Flat earth is truth, this equation is bull. Consider the solar vortext lie..
You cant equation science.
Seeing the modern angle of 89.5 degrees between Moon and sun when the former is at quarter phase shows how large an error that can result when *visually* adjudging the moment of quarter phase. The Moon traverses on the sky it's own apparent diameter (1/2 degree) every 2 hours. That 1/2 degree is the departure from 90 that occurs at quarter phase.
I would challenge anyone to reliably visually identify the moment of quarter phase to within 4 hours (1 degree), let alone rather less than 1 hour in order that the ratio dM/S be meaningful near to reality.
I agree, the practicalities of this approach are problematic. However, what fascinates me is the vision. The fact that it might be possible, in principle, to determine the distance to the Sun through a series of seemingly unrelated observations is incredible. At least it is to me
More importantly, EVEN if you were off by a factor of ten, you'd still be able to calculate that the moon and the sun are "extremely" far away form the earth. Much too far away to ever "fly" to, even if there was air all the way to the sun.
Imagine the concept where you think the sun is just up there, some where close enough to fly to, if you had wings and then get the value of 15 million Km. Even though it's the wrong answer by a large margin (by a factor of ten), it's still !impossibly! far away.
Glenn LeDrew
The modern angle is 89.853° which requires even more precision. The input of 89.5° would return a proportion of ~114 which is way off the accepted ~390. The practical difficulty in doing this measurement precisely and accurately demonstrates why the antiquated measurement of 87° was so far off, producing a proportion of just 19. Nevertheless, the genius is in recognizing the train of thought, connecting each proportion with a subsequent proportion until everything can be related to some mundane scale such as a meter stick.
Ideally, they would have taken the quarter-moon measurement over several cycles and used the mean.
The point of the video isn't precision
Great video. I learned about Aristarchus of Samos in Archimedes's Sand Reckoner where Archimedes promotes Aristarchus's heliocentric model. It would be nice to see similar videos made regarding some of the profound results of Archimedes.
Glad you liked it, thanks for taking the time to leave a comment, I really appreciate it
7:46
Ah yes, the first derivative of s with respect to m. Cool.
That was also my first thought when he introduced the notations in the beginning XD
Aristarchus was also the first (known) scientist who proposed the heliocentric model, which Copernicus was fully aware of. Aristarchus' book on the topic is not preserved, but is referenced in length by Archimedes.
Problem is, they didn´t use ellipses, and so it was still pretty complicated and didn´t really make more accurate predictions of the Solar System, and this would be even more of a challenge to Aristarchus who would have no telescopes to show other experimental evidence like the moons of Jupiter or phases of Venus which would be big nails in the coffin of geocentrism.
@@robertjarman3703 Aristarchus' model was brave and made _philosophical_ sense because it was simpler than the geocentric model. We are not talking about science in a modern sense here, with observations and "exact predictions" and so on as absolute foundation.
@@Stroheim333 I know the error is small given the imprecisions in basic measurements but you have to prove your ideas to the scientific community for them to be accepted. He couldn't provide predictions without the discoveries we would make by telescope that would have been more accurate had he also believed that orbits were ellipses, as Copernicus did.
@@robertjarman3703 As I said, his philosophy was philosophically sound, but of course he could not prove his ideas to the _scientific_ standard and criterias of _modern_ times. There existed no science in the ancient world, only philosophy.
Arístarchos also showed, as in this it could have also been deducted, the sizes of the Sun and the Moon Diameters compared... Extremely well made and calculated!
I was surprised by the quality of the video! thank you so much for posting this video.
Beautifully demonstrated and a pleasure to watch.
Thanks for the feedback, much appreciated
Great presentation; reminded me of the wonderful Dover book series on ancient Greek mathematics. Starting with observations available to all, the best natural starting point for an epistemologically grounded (rather than resorting to appeals to authority e.g.) engagement with the 'flat-earthers'. Incidentally, I have a kind of strange hypothesis, that humans can indeed detect the the very subtle but powerful curvature of the earth when high on a mountain or with a good view of a sea horizon, it just cannot be measured mechanically. I suppose that humans can sense it intuitively the brain sees it; it's obvious but very subtle, and cannot be measured with mechanical devices, but it should be possible with high precision optics and digital processing to measure out over a wide some curvature in the horizon, whether on an ocean vista, or better, from high altitude in the mountains.
Thanks for the feedback, and for taking the time to comment. Very much appreciated
With regard to your statement that we can sense curvature, this would not surprise me. We have the biological hardware to sense magnetic fields, but seemingly not the software.
Arguing with flat earthers should ALWAYS be done from first principals if you want to have any hope of succeeding. Another great argument is the Foucault pendulum, which anyone can build, and provides hands-on evidence of the earth's rapid rotation.
@@cerebralm My favorite argument against flat-earth theory is jet lag.
@@kirkhamandy
If they want to just use their eyes, let them build a water level by getting a few metres of transparent flexible plastic tube, shaping into a wide U and filling it partially with water.
Then they can check whether the sea horizon stays always at the water level when observed at different altitudes. They can even look for locations at an altitude a bit higher than where they stand, 10 or 20 km away and check whether they still show up above the water level.
One can't get much more low-tech than using a water level :-)
Brilliant, brilliant & brilliant. It's really the first time I have fully understood the maths. Thank you
Very clearly explained-well done, narrator!!
A LEGEND IS BORN!
Reborn!!
Great video - I like the way you set down the strategy for the large calculations to follow. I have always just loved how these ancient Greek geniuses did all that stuff. Of course the Earth could be flat (Eratosthenes shadow-stick) if the sun was a lot closer so that light was not reaching the earth in parallel rays, but I gather that Aristarchus did the bit about the earth-sun distance first so I reckoned he felt himself to be on safe ground - bravo. Without searching too hard, I am wondering if the "cosmic coincidence" of the moon and sun having almost identical angular sizes from Earth (hence the total eclipse calculation) could be replaced by something else? .. I suspect there are alternatives though not as neat!
Thanks for the comment, much appreciated. I am constantly astounded by the work of the Greeks, in particular the combination of simple observations with far reaching consequences. Amazing stuff
UK's science museum described this as: "a coincidence" !!! to keep the sheeple blind.
heliwave.com/Quran.and.the.Moon.html
Leicester National Science Museum if anyone cases to double check. That was in August 2001.
Please make more physics derivations. I loooove your style most of anyone else on UA-cam.
Brilliant and very clear! Thank you.
Great video
As a surveyor , I want to make a video actually measuring the angle of Moon-Earth-Sun with a theodolite someday
It’s hard.
@@larryscott3982 The actual process is simple enough. I've measured the angle before. It's the recording and "youtubing" part that's unknown for me
Di Ts
Given the differences in declination, elevation and different apparent speed of each, it’s a bitch to simultaneously instantaneously measure the subtended angle. It’s a tough thought experiment to figure out how to do it with significant, meaningful accuracy.
Di Ts
I’m thinking 2 theodolites, and call an audible for sync.
@@larryscott3982 Granted, the difficulty depends on the accuracy one wishes to achieve. You'd have to measure time for the best results. But I was thinking a crude estimation of the angle, much like Aristarchus must have done
What an excellent way to spend 25 minutes. Thanks man!!!!
Nice work! Also we must note the fortunate coincidence that the disks of the two bodies seem from our perspective almost exactly equal.
Deeply inspiring.
What the human brain is capable of.
Fantasy.
Very cool demo! I would have liked to have seen you use the actual observational values of the Greeks (87°, 2 instead of 2.6, etc), and then compare to the known values of today. Maybe even then compare to the Greek method using modern values, vs. values we've obtained other ways such as through the Apollo reflector experiments.
Also, one question about the methodology: did the Greeks know trigonometry? If so, how did they calculate values for sine and cosine? I'm almost completely certain they didn't have infinite power series back then.
Excellent. The power of trigonometry and basic geometry!
First, this was beautiful and enjoyable, so thank you for the great video. Obviously the calculations and diagrams are 2D cross-sections of the actual 3D situation, which leads me to ask if you could elaborate on how the location of observations of Earth enters the full calculation. Also, a video like this on the tides would be brilliant. Thank you.
03:04 How did he get a mesurement for theta?
10:10 And how did ancient greek astronomers find the value of 2 (or 2.6) for the ratio Rshadow over Rmoon?
When the Moon is exactly half, you have both the Moon and the Sun in the sky at the same time. It is easy to sight a line towards each of them from a fixed point and measure the angle between those two lines.
I don't know how the ancient greek found the value of Rshadow. The way I would do it is to wait for a lunar eclipse, time how long it takes for the shadow to cover the Moon, how long totality is, and how long until the Moon is fully out of the solid shadow. The solid shadow is called umbra, it is surrounded by a dim but not completely dark area called the penumbra. What I would measure is the umbra, in terms of time. If it takes x seconds to cover the Moon, and the time from the shadow started to cover the Moon until the shadow has completely cleared the Moon again is 3x seconds, then I know the shadow diameter is 2 times the Moon diameter.
Another way, which is difficult with naked eye observattion but easy with a telescope and a camera, is to take pictures while the edge of the shadow moves across the Moon. You will see the shadow is a partial circle. Find the center of that circle and measure the radius.
@heart momentum "Can you explain with equations?" I'm not sure I can do that in a way that adds much clarity without a drawing. Let me offer a different example, that might make it easier to see how it works.
Imagine you have a car that is 10 meters long (like a small truck). You are about to drive across a bridge that is 20 meters long. Taking time as zero when the front of your car is at the beginning of the bridge, it takes X seconds until the end of your car is also on the bridge. After another X seconds, the front of your car is at the end of the bridge. Yet another X seconds later, the end of your car is at the end of the bridge. Thus, 3 times X seconds to completely cross the bridge. The bridge is twice as long as your car. Here, the car is the Moon and the bridge is the Earth shadow.
In a formula, a is the length of your car, b is the length of the bridge, x is the number of seconds to completely cross. Then you have b = a*(x-1). The -1 comes from the fact that you have traveled an extra car length while measuring the time to cross. Consider the back of your car; when you start the timer it is a whole car length behind the beginning of the bridge. So when you measure how many seconds to clear the bridge, you have actually measured the time to drive the bridge length plus the car length. That's why I originally mentioned 3x instead of 2x which would be more intuitive, but wrong.
I am not sure this helps, but at least I tried :)
@heart momentum "May I ask how one can view the Sun in a night sky?"
- If you need to do that, you can measure the rate at which the Sun moves across the sky, and extrapolate from sunset. But you don't need to do that, because at half Moon the angular distance between the Sun and the Moon is close to 90 degrees, so they will both be in the sky together. The sky from horizon to horizon is 180 degrees.
@heart momentum Right, I was not too clear on that. a and b are the lengths of the car and bridge in this example. But when observing the Moon and the shadow from Earth, they would be the angular sizes instead. The shadow size is b, the unknown. You can measure a, the angular diameter of the Moon at a full Moon, so not the same day as you measure the angle between the Moon and the Sun. The time x (which I should probably have called t instead) can be measured in seconds for example.
@heart momentum there is no reason why this has to be done at night. the sun and moon regularly appear in the sky together.
Great video!
One thing though: @4:48 Why do we assume that the tangents of moon and sun meet at the center of the earth. The two lines could meet at any point. The fact that we see a shadow on the surface of the earth means that they dont meet before touching the surface of the earth. But that doesn't mean they meet in the center of the earth. In fact it would be entirely possible they never meet (Iin the case that R_s = R_m). This observation means that the two triangles are NOT similar.
Exactly
it has been pointed out elsewere in the comment section but just in case you're still wondering you can assume that the angular size of the sun and moon are aproximately similar by the fact that there are annular and total solar eclipses (in annular ones the angular size of the sun would be slighty greater and the opposite would be true for total ones), if the angular size is the same the tangents would meet in the retina of your eye but given the distances we're mesuring here the distance between your eye and the center of the earth is negligible.
AT THE OUTSET, THIS IS A GREAT VIDEO ..... WELL EXPLAINED !
NOW A LITTLE CORRECTION:
TRIGNOMETRY WAS NOT IN VOGUE DURING ARISTARCHUS TIME. SO, HE PROBABLY USED THE LAW OF SIMILAR TRIANGLES POPULARISED BY EUCLID (WHO LIVED AROUND THE SAME TIME)
AND WHAT HE DERIVED WAS ARGUABLY A RATIO OF THE DISTANCE BETWEEN EARTH - MOON AND EARTH-SUN. ACCORDING TO MOST SOURCES THAT RATIO WAS 1:20. TODAY WE KNOW THAT RATIO TO BE 1:400
PRETTY REMARKABLE AT THE TIME, AS TO THE MORTALS WHO WOULD GAZE AT THE SKY .... SUN AND MOON WERE OF SIMILAR SIZE AND MANY WOULD HAVE ASSUMED THEM TO BE EQUIDISTANT TO EARTH.
Top, really cool. Enjoyed it
Totally awesome video
Glad you enjoyed it
Thanks again … enjoyable channel ...
Cheers!
Basic geometry but still so many steps, great walkthrough!
really nice :- ) thanks, another fun calculation is the radius of the earth by Al-Biruni
Given the huge span of creativity and vision that lies behind these calculations, one is left wondering how they squared all this with Ptolemy and the geocentric view of the universe. Aristarchus and Eratosthenes (among others) were real thinking people seeking the truth. They must have seen the flaws in the epicycle theory and sought an alternative mechanism .. were they really the first Copernicans I wonder? we'll never know.
In part 2, it seems that a necessary part of the argument being made is that the shadow formed during a total solar eclipse on the earth's surface comes to a focus exactly in the center of the earth. The way I see it, imagine the earth being half the distance it currently is to the moon. The focus would be at some point beyond the earth and yet there would still be an eclipse on the earth. How did Aristarchus justify that the focus was at the center of the earth during a total eclipse?
I am having problems mentally purchasing part 2 for exactly the reason you describe. It seems like an unlikely coincidence. Another thing I find difficult to accept in part 2 is that there is the assumption that there are right angles between the line that goes from the center of the Earth to the center of the Sun and the radii that meet the tangents to the Moon and Sun. But we know from geometry that tangents to a circle meet the radius at a right angle. It would seem we have triangles with two right angles? This doesn't add up in my mind. Something is wrong.
That is true, the good approach would be to say the rays converge at earth's surface, as it is from there where you are observing both bodies to have the same apparent size.
@@patricklincoln5942 The reason why the tangent of the moon and sun isn't at 90 degrees to the radius in the diagram is because the radius when it is straight up is a good enough approximation to the radius that touches that tangent line so it doesn't really matter in the calculations.
Because in solar eclipse, you are able see little of the sun around the moon (I mean that gas thing around the sun). That means it is quite close to being in focus on surface of earth. And it is quite close being in focus on center of earth. So it is good enough approximate.
In a solar eclipse you can observe that the sun and moon overlap almost exactly. All that remains visible is the corona. That implies the focus is the center of the Earth. If the moon were smaller or further away you'd see extra sun; if it were closer or larger you'd see nothing at all. (In both cases you could determine their relative sizes like with the lunar eclipse)
This is indeed an amazing coincidence, but it's really true!
Excellent presentation and explanation. Nice work!
Thanks!
I really like this video; it clearly explains the concepts behind the calculations, as well as the calculations themselves.
My one complaint is it doesn't explain how the measurements were made. The modern numbers are just assumed to come from on high and plugged into the derived formulas.
Makes you proud of human antiquity. Human always searched for truth and knowledge. We got stuff wrong along the way but we keep trying
If this channel doesn't become the primary point of reference for final year high school / first year undergraduate physics, there is something wrong with the world. The maths makes my head hurt, but the argument is as clear as crystal...
THIS CHANNEL IS FULL OF UNACCEPTABLE ASSUMPTIONS
Try it and see how close to a big fat ZERO you will get.
YES this is a good youtube video, not like the others that are minute after minute of flashy docuscience pictures for ADD kids.. within seconds you're actually TELLING ME STUFF I DIDNT KNOW
This is an excellent video, well done. I will be sharing this with anyone who asks how we know how are away the sun and moon are.
Amazing!
Thanks!
Awesome video......
Thanks!
@@PhysicsExplainedVideos expecting more such videos
I thank you for the effort put into this. While some may complain about the form, I think it was well chosen. As long as there is people able to make and understand these observations, humanity is not lost to the anti intellectual insurgence (FE et al.) and the children of my children can still hope to reach above and beyond our current limits.
I knew that the greeks calculated the diameter of the earth. I had no idea that they were able to take this information to derive the distance of the earth to the moon and the earth to the sun. This is wonderful.
4:30 "And if you put this into a calculator...." I am sure that is what Aristarchus did to solve cosine functions :).
Actually, Aritarchus didn't use trigonometry in his calculation. He used algebra to deduce ratios of objects.
4:26
@@manofgod7622
😂🤣😂🤣😂🤣
It is the simplest possible arithmetic. If I can do it without a calculator, I am sure he could.
Because we're dealing with 1/cos, the slightest change in that degree makes huge changes. 1/cos(89) is so much different than 1/cos(89.8)
Interesting, Euler and his Cosine function apparently were used in Ancient Greek calculations...
They were used, but with another way. Greeks were using a word-like calculus, think it like modern calculus described in words only.
Apparently, cosine and sine functions were used in ancient Greece, since Pythagoras' time.
There is a clue: sine in Greek is ημίτονο
Ahh; interesting observations. I see someone said 'calculus with words'. It was discovered in the 90s maybe that Archimedes was doing calculus with geometry basically. Fun stuff.
Wonderful video! One question - in the first explanation how was theta, the angle between the lines connecting the earth to the moon and sun, measured? How was it estimated to be 87 degrees at the time?
A great thanks for this brillant represantation. Indeed you are excellent.
You're very welcome
One thing missing - what was the original values that were used and what results were obtained?
You make an interesting point. The number system used by Aristarchus was not the modern decimal system used here but what we now call Roman numerals. pi was not represented by 3.14159 . . . but, according to Archimedes, the Roman numeral equivalent of 223/71< pi < 22/7. I don't know the procedures used but I imagine multiplying 390 by 22 and then dividing the result by 7 (CCCXC times XXII divided by VII) would be quite a challenge.
@@professorsogol5824 roman numbers for greek calculations lol you do realize that greek had their own numeral system
and hint they used the letters of the alphabet as numbers
www.britannica.com/biography/Aristarchus-of-Samos
@@sonaruo I stand corrected. In my previous effort to look into this question, I read someplace that Roman Numerals actually pre-date the Romans. But I also see that from about 400 BCE the Greek system was not a decimal system but an additive system where there were 9 unique symbols for the numbers 1 to 9, 9 more unique sysbols for the numbers 10 to 90 and 9 more unique symbols for 100 to 900. But my basic question still stands: How did they add, subtract, multiply and divide? What are steps needed to add (222) and (589)?
edit: I see UA-cam did not reproduce my question as written. I cut and pasted the ancient Greek symbols for 2, 5, 8 and 9 to express the two numbers in the ancient Greek notation. However, these symbols were lost. Let us re-define the Greek numerals as follows: A is 1, B is 2, etc I is 9, J is 10, K is 20, etc, S is 100, K is 200, etc (This system will break down soon as there are not enough uppercase letters in the Latin alphabet, but the system will allow us to express the problem above) Thus the problem above becomes TKB + WQI (according to the above transliteration of the ancient Greek system TKB represents 200 + 20 + 2, or 222 in the modern decimal system). How do we re-write, re-arrange and determine that B + I = JA? How do we carry value 10 (J) into the next step?
@@professorsogol5824 well now what you mean by decimal because Greeks also is based 10 system the only difference is how they represent the digits
instead of a 10 digits that repeat they simple had more digits to do the same job
in greek the 2 number you wrote will still have 3 digits and they will add up as you and me no problem since base 10
and they went more since they used symbols next to the letter to introduce even bigger numbers
think as if they had a different symbol for 9 depending on where the nine was and the same was true for any other digit
...and whereupon seeing Aristarchus completing his calculations to such a degree of accuracy, and learning for himself the vast gulf between Earth and her most apparent neighbours, the mighty god Apollo threw up his hands at the ingenuity of humans and said, "Alas! They shall never name a space program after me."
And Ares just laughed and replied, "My beautiful friend, just wait until they try coming to MY house. Though they may possess Curiosity, they shall never have Opportunity."
Best comment so far
I've always thought, that since so much history and ancient knowledge is lost through time, I bet there were dozens (hundreds?) of other times through human history that someone was smart and inventive enough to figure this out. That Aristarchus was not the 'only' one, he was just lucky enough to have his findings persist through history so that we can marvel at the intelligence of ancient man.
Merci Physics Explained, i will be able to answer my friend who is dubitative modern scientist could achieve to measure that kind of distances. "Ancients greeks could do this by watching the sky ans counting their paces, using beautifull maths and..a stick" thanks you so much, definitly love your channel
8:58 don't write =. Use => or to show that one follows from the other
To make a man walk 800km because of a stick makes one wonder 'the carrot'? This should be compulsory viewing at all Flat Earth Conferences :-)
I totally agree!
@@PhysicsExplainedVideos Today we can duplicate that with cellphone and a buddy somewhere else... Science was a LOT of work back then.
Just saw a video detailing the trouble that Max Planck dealt with when coming up with the quantization of EM energy. He toiled for years.
@@billallen275 Indeed Imagine the poor anonymous bastard trying to walk 800 km in a straight line, measuring any step. His accuracy allowed such a breakthrough, it is still celebrated 2200 years later.
@@FranFerioli
I'd like to read the account of that. Guess it's documented.
@@billallen275 It is not documented. it is a story of a fictional character There is no geometric horizon as it is never been observed. How did Al Biruni measure something he could not observe?
Excellent explanation. thank you.
thanks. great work. well explained
I didnt understand a thing. But I highly recommend his pen, it's really good.
It really is just some basic high school math. But it is amazing how creative the Greeks were to put these things together and derive these maths.
Also, what pen is it? Asking for a friend ;D
Nice handwriting, but it’s odd to me that he holds the pen correctly until he starts writing.
I was thinking the same thing. He holds the pen very strangely when writing.
They don't teach writing anymore. LOL
GordoGambler Or it was taught but the person here didn’t follow instructions. I was taught properly, but by high school I had developed an incorrect grasp and my mom had to reteach me how to hold a pen.
Great video mate, very nice explaination. But can u pls elaborate on how different theta and Rshadow must have been calculated. Thanks
Wonderful explanation. Well done!...
flat earther: ewwww math
you can see with the naked eye the crater called "tycos" on the moon... if you believe you can see something 70 miles wide from 238k miles away then you need this math to ease your cognitive dissonance. Earth is flat.
Do not believe, do the maths. It's like barely seeing a person't thumb at 50m = 160ft.
@@adriangourlay Was it your friend Dunning who told you this?
You can NOT see the Tycos crater on the moon with the naked eye. What you CAN see, is a white spot that is a LOT larger than the crater.
This has been proven, and can easily be proven.
So basically, you claim that we can see something on the Moon that science tells us is 70 miles wide.
This is wrong. What you can see is not 70 miles wide. And science does not say it is 70 miles wide. What science tells us is 70 miles wide, is something else.
@@adriangourlay I'm tired of flat earthers making up random stuff to say that the earth is flat, how about you guys explain how the sun even sets on a flat earth while being visible to others without your ad hoc excuses?
Those calculations only work assuming the earth is round. When you use the Scientific method, you will see that the proofs show a flat earth. Run a a laser beam across a frozen lake hundreds of miles across at a fixed height horizontally and it will maintain the same height. At 25,000 mile circumference, that would be 7.96 inches per mile squared. 100 miles equals almost 80,000 inches in drop. No such drop is observable or measurable. Science rules!
Flat earthers be so confused
Excellent video. Clear presentation.
Absolutely Superb!
Alright, I will bite: Where did the Greeks get the trig tables?
Their mind. They could complete trig tables using any circle. It's not a big deal.
en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords
They invented trig tables. DIY trig tables.
Also, with our modern understanding we like to think in terms of degrees but there’s a good chance early geometry was simply left as ratios, like radians. Meaning you inherently communicate some triangle when describing an angle. For example, sin(1/root2) inherently communicates a 1:1 right triangle thus the angle is 45°. Or sin(1/2) is a 1:2:root3 triangle and so it’s 30°.
Sphaera Invictus
Right. The sine(45deg) or 100 grads, or 1600 Mils, or 1/4(pi) radians, 0.25 Tau is the same. I can make up an angle unit. It all comes back to fraction of a circle
To the author of this video: I have a big problem with part 2. You assume you have right triangles where the right angles are along the line segment that stretches from the Earth's center to the Sun's center. However you also assume (implicitly from the diagram) that the other line segments are tangent to the Moon and the Sun. We know from geometry that the radius of a circle meets a tangent to a circle at a right angle. It would seem that your right triangles have two right angles. This does not add up in my mind and I fear that there is something wrong with your construction. I also think it is far from obvious why the shadow should just happen to converge at the center of the Earth. The arguments you make depend on this fact (I am not even sure that is the case...it seems to be an unlikely coincidence). I suspect you have misunderstood the assumptions associated with your source. I suggest you take another look at it where you give your analysis of your source a more rigorous treatment.
I think the fact that the objects in the diagram are circles is inconsequential. The visual representation of these objects as seen by an observer at the center of the Earth is a flat round disk so you could replace the circles by vertical lines in the diagram.
@@Nardypants Exactly. All this Part 2 is doing is to express that if the sun is 390 times further than the moon (the actual number is ~400 times), and if the apparent angle of the moon and sun are the same, then the radii of each (or their diameters) must have the same proportion. There's no problem there.
I find more issue with the assumption of equal angular size of both bodies...for three reasons. First, you aren't literally standing at the center of the earth, so your surface observation contains a degree of measurable error (about 4000 miles worth). Second, in a total solar eclipse, the moon's apparent angle can be slightly larger than that of the sun...it's not an exacting measure, especially when you consider where you are standing. Third, these measures change depending on where the moon is in its orbit relative to the earth...apogee vs. perigee. This means the apparent size of the moon varies as much as ~14% depending on where it is in its orbit.
This is the easiest-to-follow explanation of the work of Aristartcus of Samos I have ever seen. Nothing is left to chance in this video. Ñ
Very nice. I wish more people would take interest in this.
The Earth is flat.
Just kidding :)
He should have just Googled it
Excellent presentation ! Clear and the maths should be simple enough for most students.
Thanks for the kind words, much appreciated
@Tony Ortega
It is explained in the video. When the a Quarter Moon ( 1st or 3rd) is observed it forms a right triangle Sun-Moon -Earth . The modern angular measure from the Moon to the Sun from the Earth’s reference is about 89.85 degrees .
So a relationship based on the distance of the Moon-Earth = 1 therefore by taking the tangent of 89.85 one gets that SUN is 400 times farther from the Earth than the Earth is from the Moon .
@professeur essef
As previously posted other ancients had better technology and got better results ,his just lasted in the history books.
listverse.com/2015/09/13/10-incredible-astronomical-instruments-that-existed-before-galileo/
@professeur essef
It appears that you have trapped yourself into a pseudo religious philosophy. This is simple observations of the Moon Earth and Sun which sets up a series on similar triangles. Once the series of ratios is calculated then it one can then put actual measurements in place . How accurate are those measurements that is dependent upon the technology at the time but maths hold up .
Review the video take screen shots of the diagrams and work it out yourself so you can see the logic . It just maths .
Very nicely done, I purchased the book to learn more and will make a few videos myself.
Add this to it if you seek the Truth:
Please see how chapter The Moon of the Quran gives the min/max distance to the Moon as 102/117 moon diameters. heliwave.com/Quran.and.the.Moon.html
Nicely explained, for a preset of set numbers.
Interesting video.Thanks.
tks so much, well explained for a physics strained person as me
Great video :D I lecture on pre-socratics and post-Plato's Academy tied thinkers, and I always mention Eratosthenes (who was a librarian in the Library of Alexandria, among other things). Trigonometry had been used in calculating physical phenomena at least since Anaximander, who is often argued to have first applied it to the solar clock (shadows of a stick, used to present time).
You are saying that they had access to trig tables?
@@bizopca It was before Pythagoras, so the calculations would be about length of the shadow cast on the ground, so as to estimate how far into the day it was. Likewise, Thales was said to have estimated the height of the great Pyramid, by comparing its own shadow to that of a stick (Thales theorem is about
analogies).
@@thephilosophyofhorror But, the method we just watched can not be how the Greeks did the calculation, right?
@@bizopca Which calculation do you mean? :) Pythagoras (or whoever of the pythagoreans did it) came up with his theorem in the same period, and after that you'd have all sorts of trigonometrical calculations. Aristarchus lived centuries later.
michael webster
They invented trig tables. Sine is calculated.
Great video! One quick question: What are the MINIMUM and MAXIMUM lengths of the shadow cast by the Moon, based on the Moon's closest proximity to the Sun (i.e. a New Moon during perihelion) and its furthest proximity from the Sun (i.e. a Full Moon during aphelion)?
One of the most important thing from the video is Not that ancients Greeks could determine distances between Sun Moon and Earth using sticks on the sand but that they understood the concept - Earth is NOT FLAT!! Hello from 21 century
Great! Thank you so much! 👍🖖🍀
truly amazing what math can do for us!