Move over, Newton and Leibniz. Archimedes may have beat you by 2,000 years:
Two of the texts hiding in the prayer book have not appeared in any other copy of Archimedes’s work, so no one but Heiberg had studied them until now. One of them, titled The Method, has special historical significance. It could be considered the earliest known work on calculus. […]
The Greek philosopher Aristotle built defenses against infinity’s vexing qualities by distinguishing between the “potential infinite” and the “actual infinite.” An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn’t exist.
Archimedes developed rigorous methods of dealing with infinity—still used today—in which he followed Aristotle’s injunction. For example, Archimedes proved that the area of a section of a parabola is four-thirds the area of the triangle inside it (shown in red in the diagram below). To do so, he built a straight-lined figure that’s an approximation of the curvy one. Then he showed that he could make the approximation as close as anyone could ever demand to both the section of the parabola and to four-thirds the area of the triangle.
The writings had been hiding in plain sight, in a palimpsest underneath a book of prayer. One wonders about the monk who scraped the parchment clean. Did he have any idea? Could he have…?
The postulation of a lost age, where human beings had made great advances in science, medicine, and mathematics, has always made for wondrous fiction (especially when the ancients had steampunk spaceships.) This discovery leaves me despairing at the fragility of progress.
The Edge of the American West 
30 comments
January 25, 2009 at 10:16 am
jay boilswater
The Antikythera machine, as another example.
January 25, 2009 at 10:29 am
andrew
So is it “calculus” or “the calculus”? I wonder if Archimedes could settle that one.
January 25, 2009 at 10:43 am
saintneko
Well, we know China had chrome technology 2000 years before us. We know someone had fantastical clockwork machinery hundreds or thousands of years before we had clockwork hackers near the time of the Renaissance (feeling too lazy to wikipedia my story, coffee grinder is still drying out and until I’ve had my jet fuel…) – we know the Mayans calculated the motions of the heavens pretty much down to the second through the next few years, using some rocks and berries (okay I have no idea what they used for non-stone writing material).
Anyway, my point is, the amount of knowledge lost of the years is probably a staggering amount. Who knows how many patent clerks through the years thought up great inventions only to have war or some dumb ass who needed scrap paper erase that knowledge, potentially forever. And that’s just the non-intentionally malicious effects. Who knows how much knowledge has been suppressed with malice aforethought.
Kinda makes you wonder about rumors like Atlantis though, knowing how easy it is to lose something as world-shattering as calculus for 2 thousand years (no calculus, no computers, no space exploration, etc etc etc etc). Also makes you wonder if maybe civilizations haven’t risen and fallen with other intelligent animals through the hundreds of millions of years prior to us, just they never got beyond the wood-working stage so no remnants remain. Or maybe they did, and their instruments were both A) too alien for us to recognize and/or B) have just decayed too much to be noticed. If dinosaurs wore clothes, would we recognize it, or would be put that texture down to ‘skin’ or ‘feathers’ ? What if they wore animal skins or feathered clothing?
NOT that I believe that about dinos, mind you, it’s just a thought exercise. But watching shows like Futurama, whenever they reference the past, leads me to believe that they may be partially right – we can guess about the past, but we’ll never know for sure until we invent time travel.
January 25, 2009 at 10:51 am
saintneko
Re: Andrew –
Much as we now have “the internets” “the intertubes” “the suck” – Newton was just, as usual, centuries ahead of his time. I think he was actually a dimensional engineer in his sleep, opening wormholes to the future and copying down the ideas and formulas mentally for his waking mind to ‘discover.’ Of course, this would mean he was reading work from the future that would exist if he hadn’t peered into the future to copy it down so someone could write it down in the future for him to peer into and then copy it down so someone could… well, you get the picture.
January 25, 2009 at 11:22 am
David Weman
It should be pointed out this isn’t something that was announced yesterday. When the article writes “until now” it means until the last ten odd years.
January 25, 2009 at 11:25 am
dana
Yeah, it’s definitely for “some value of now.”
January 25, 2009 at 11:54 am
ben
IIRC from previous discussions of this text, what Archimedes was doing wasn’t really calculus, like not really really.
Plus, people have known that ancient mathematicians knew about the method of exhaustion since forever, practically; that’s how they used to calculate the value of pi. The proof is I suppose different from that, though. Sigh.
January 25, 2009 at 12:27 pm
Hemlock
Actual infinity as potential infinty fuels present debates in astrophysics. Go, go, Aristotle!
January 25, 2009 at 1:24 pm
Jason B.
I’m eagerly anticipating academic references to “teh calculus.” Heh.
January 25, 2009 at 1:51 pm
NeverEscape
…and we’re eagerly anticipating blogger explication of calculus. Heh.
January 25, 2009 at 2:08 pm
kid bitzer
ben–netz is in your neck of the woods, no?
see if you can ask him how this new calculation of volume requires us to suppose that archimedes employed an actual infinity of slices.
did archy also do any arithmetic with them? e.g. multiplying, adding, subtracting actual infinities? what does “operate with them” mean?
suppose i take a cube and slice it into two prisms, the plane of division including the diagonals of opposite faces.
then i try to calculate the volume of a prism, and reason that, in any plane normal to the plane of division, the area of the prism’s projection is one half of the area of the square’s projection. (e.g., on the side facing me, i see a square cut into two isosceles right triangles).
from this i conclude that the volume of the prism is one half the volume of the cube.
did i “operate with” actual infinities? even if, in fact, i believe that there is an infinite number of planes normal to the plane of bisection?
more and better particulars, please.
January 25, 2009 at 2:45 pm
Hemlock
Perhaps I can demonstrate a possible paradox resulting from actual and potential infinities in the context of Newton and astrophysics:
So Newton’s inverse square law posits that as two object move farther apart, the lass gravitational attraction. If you double the distance, F decreases by a factor of 4 (2 squared); triple the distance, F decreases by a factors of 9 (3 cubed); and so on. But why doesn’t it decrease by the cube of the separation? Well, a sphere has a surface area proportional to the square of its radius, which means that the density of graviton field lines (total/area) decreases as the square of separation. As you increase the distance, field lines uniformly spread out on a sphere with four times the surface area and the gravitional pull will drop by a factor of four.
This works only if there are only two spacetime dimensions, cuz circumference is proportional to the radius, and the F will drop by two (instead of four). Hence there must be more, which is fueling the debate between positing actual infinity as potentional infinity (proof necessary). I can go into the final statement if necessary, but that’s a good example.
January 25, 2009 at 2:46 pm
Hemlock
This is all rudimentary stuff–yes, stuff. Heh.
January 25, 2009 at 2:49 pm
Hemlock
Corrections: this works only if there are MORE than two spacetime dimensions. Hence Planck length 10 ^33 comes into play. Grandfather used to be an engineer/physicist, still likes to ruminate on these issues when his grandchildren visit.
January 25, 2009 at 2:55 pm
Ahistoricality
It seems to me that there’s a kind of elastic limit on progress at any given moment in human history. The calculus and heliocentric cosmology couldn’t take root in the Helenistic world because the rest of the technology and cosmology had no use for them (I know, there were uses, but not wide enough to get a strong handhold, is my point).
There’s a sort of “marketplace” effect, whereby ideas that are too advanced fail because nobody else can figure out what to do with them.
January 25, 2009 at 2:59 pm
Hemlock
Correction: 10^-33. Heh. 10^33, and we’d all be in big trouble. Heh.
January 25, 2009 at 4:33 pm
Doctor Science
Ahistoricality:
I agree with you completely. And “what to do with” an idea doesn’t necessarily mean “how to use it to make money”, it may mean “whether it’s worth the trouble of thinking about.” Thinking new ideas and making new things is hard, it takes time and effort, so people won’t generally do it unless there’s some payoff.
January 25, 2009 at 5:41 pm
Cosma
here is a nice piece by Netz from 2000, where he cautiously says that
Of course things may have changed since then; I haven’t read Netz’s book on Archimedes.
January 25, 2009 at 5:50 pm
Cosma
Duh, there’s a website for the palimpset project. There’s a little snippet by Netz on “Methods of infinity”, where he seems to address kid bitzer’s last question:
The last bit seems like an overly dramatic way of saying we don’t know whether the continuum hypothesis is true, i.e., whether there are more real numbers than there are sets of integers.
January 25, 2009 at 6:16 pm
Cosma
despairing at the fragility of progress
On which note, let me share the most astonishing paragraph I read last year (p. 75 of Barry Allen’s (very uneven) Knowledge and Civilization:
January 25, 2009 at 6:23 pm
teofilo
we know the Mayans calculated the motions of the heavens pretty much down to the second through the next few years, using some rocks and berries (okay I have no idea what they used for non-stone writing material).
Paper made of bark. I just read a (good) book about this.
January 25, 2009 at 6:24 pm
kid bitzer
this is actually more useful than netz’s presentation:
http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/Archimedes%20Method/Prop14/Arch.Method.Prop.14.html
no “operations” on, in the sense of addition, multiplication, etc.
but there is the assumption that two given infinite sets are equinumerous.
yes, that does look like an ‘overly dramatic’ expression of the continuum hypothesis. but then netz’s favorite curve seems to be the hyperbole.
January 25, 2009 at 6:26 pm
teofilo
On a more general note, one thing I’ve found pretty remarkable as I read more and more archaeology is just how much we don’t know and have no way of knowing.
January 25, 2009 at 6:26 pm
kid bitzer
sorry, my “that” jumps backward to cosma’s paragraph, not to my immediately preceding paragraph.
January 25, 2009 at 8:18 pm
Kieran
The Antikythera machine, as another example.
A few times over the past view years when teaching social theory, I’ve mentioned the Antikythera mechanism in passing, in the course of talking about technology and social change. The same thing always happens: no-one’s heard of it; I give a quick sketch of the story; and then they are completely fascinated and talking about it takes up the rest of the class period.
January 25, 2009 at 8:52 pm
ben
but then netz’s favorite curve seems to be the hyperbole.
Zing!
January 26, 2009 at 1:19 am
michael holloway
And perhaps, all this goes to show the history of progress is non-lineal – just like everything else.
-great links.
michael holloway
January 26, 2009 at 6:09 am
Prof Burgos
Again the importance of paper — physical stuff — comes through loud and clear.
Pity the historians – social, diplomatic, military, art — of the future. Apart from mandatory government backups, what will become of the concept of the “papers,” stored for future researchers at Libraries of Congress, Archives of Nations, and the like?
“For my dissertation I will draw on the hard drives of Joe the Plumber….”
January 26, 2009 at 7:27 am
Michael Turner
Maybe I’m not taking Prof Burgos’ point as intended, but it’s not just the importance of “physical stuff”, it’s also how easily produced the physical stuff is. Paper was mass-produceable, which made knowledge mass-distributable. Palimpsests were the preferred mode of recycling for a while because parchment was the only technology for making a book, and parchment was expensive. If parchment has a virtue as a medium, it’s that it’s less perishable than (acidic) paper.
At some tipping point in the economics of data replication, the problem becomes one of indexing everything, not of losing much of anything. 20 years ago the idea of backing up your hard drive to a generic service bureau was a dream. About 10 years ago it started becoming a reality. Another 10 years from now, versions of Windows might come with that option enabled by default.
While I’m at it: I’m not sure I’m down with this theory that how much knowledge matters, in practical terms, is a major factor in its survival and propagation, when it comes to mathematical concepts. Other factors — is the theory accessible to enough minds? is it likely to purged as heretical under some ideological regime? — might loom larger. Calculus is, intellectually, a bit of reach for most. (Isaac Asimov once confessed that he majored in chemistry because physics turned out to require calculus, a subject that defeated him. Say what you want about the guy, he was no dummy.) And the notion of infinity is open to all sorts of theological mulling, much of it productive of very little except bitter controversy. It might have been just such controversy that Newton (also a theologian) might have been avoiding by recasting results derived by differential calculus into more pedestrian algebraic forms for publication.
The utility-leads-to-propagation theory might also be turned on its head in the case of Newton: maybe he just wanted to keep his techniques a secret. Usefulness can breed a proprietary attitude, which can lead to “lost arts”, if carried too far.
January 26, 2009 at 12:51 pm
Prof Burgos
I was actually thinking more in terms of the physical act of research, the fact that someone was physically able to go onto the paper and find the ghostly “something,” in much the same way that one finds paintings beneath paintings in the Old Masters, or can follow Hemingway’s (or Joyce’s or Whomever’s) revisions to a manuscript by actually handling the actual manuscript in archival research.
The virtuality of data — of knowledge — renders that kind of research far more problematic. When I was rummaging through files of the Near East Division at State from the 1940s, it was interesting to read the marginalia on memoranda and position papers. Now much of that has disappeared — sure, there’s mandatory government archiving, but the research task will be much more challenging. And there’s no archiving requirement, of course, for private citizens.
I go through my own e-mail and come across these infinite iterations of one- and two-word Re’s to Re’s to Re’s to FW’s to Re’s. And even though these comprise a kind of marginalia I suppose, they lack a necessary human element that one sees in double-underlining, or heavy pen pressure, or whatnot.
Also problematic is the question of data standards — I quickly abandoned a once-and-future-research project because the data were all on those big tape drives from room-sized IBMs in the 1950s and 1960s — the cost of conversion was astronomical and, given the uncertainty attendant to archival research, prohibitive. I went in the Army in the early 1980s and left in the early 2000s and in that time went from a proprietary UNIX-based system to DOS to every iteration of Windows. All of that data was stored — but stored in its original form — and when I recently tried to open a file I had from ‘way back when (on a 5-1/4″ floppy disk!) it was completely unintelligible.
So that’s a lot of random musing about what might amount to a get-off-my-lawn grampy complaint — paper matters for research.