Along with Euclid, Newton, Leibniz, Gauss, Pascal, et al., Archimedes (287 BC-212 BC) is widely considered one of history’s most brilliant mathematicians.
Were a time-traveling calculus text (combo first and second-year college level) to fall at his feet–without precalculus material included–would he likely quickly grasp the concepts, or lack the requisites and find it a tough slog? (Let’s imagine he can read the text.)
Mods: I’m hoping we can keep this reasonably factually based.
Considering he invented Calavari’s method about 2000 years pre-Calavari, I suspect he would understand at least the basics of integral calculus.
He might have some initial trouble with the notation, and he might be somewhat suspicious of the reliance on algebraic rather than geometric concepts (as indeed Kepler was). And the analytic-geometry part (graphs of functions and stuff) would probably be pretty counter-intuitive.
Certain of the pictures (solids of revolution, conic sections and their tangent lines) would probably be pretty obvious to him, though.
And SaintCad, I think you mean “Cavalieri”?
I agree, they wouldn’t understand the nomenclature. But once they got past that, they all had the sufficient backgrounds to figure out the rest. Assuming they knew it was valid, blah blah blah…
Only if it’s written in a language he understands. I don’t think an English textbook would do him any good at all.
I think we have to assume a textbook in Modern Greek, which should be close enough to his Greek to be comprehensible to him.
He would be as snowed by it as every other sharp new student, and then pick it up over the course of the semester.
Didn’t the Greeks have trouble with the concept of infinite series, as per Zeno?
Yes, as a philosophical issue. Present-day mathematicians still find that infiite series raise philosophical issues, and these issues might well be mentioned in elementary calculus texts. The issues are still there, but modern mathematicians have found ways to cope with them, and Archimedes would probably understand what’s going on.
For example, one paradox suggests that a runner can never overtake a tortoise travelling much more slowly. The ancient Greeks knew that in real life the runner does overtake the tortoise: that’s why thinking about the paradox is so interesting.
Don’t bring Lost into this, please.
Or you could assume, ** as instructed in the OP ** that he can read the language.
In mathematics, Archimedes extended the ideas of Eudoxus and made use of potential infinity in finding areas and volumes using infinitesimal quantities. By these methods, he derived the rule stating that the volume of a cone inscribed in a sphere with maximal base equals a third of the volume of the sphere. Archimedes thus showed how a potential infinity could be used to find the volume of a sphere and a cone, leading to actual results. After Archimedes’ death at the hands of a Roman soldier, a stone mason chiseled the cone inscribed in a sphere on his gravestone to commemorate what Archimedes considered his most beautiful discovery.
http://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm
Perhaps the most important insight into Archimedes mind is a passage in the Method in which he describes the concept of infinity. Infinity is a fundamental concept in all of mathematics, refined by Newton’s invention of calculus, but was previously considered a problem too difficult for ancient Greek mathematicians. From the palimpsest, we now know that infinity was understood and described by Archimedes twenty centuries before Newton.
http://www.exploratorium.edu/archimedes/peering.html
Archimedes exploits with the infinitesimal brought him to the cusp of discovering integral calculus. To find the area of a parabola, he attempted to fill the area of the parabola using rectangles and triangles. As the polygons grow smaller, the approximation becomes more accurate. This is a concept that leads to integral calculus. In calculus, the rectangles used to sum the area of the parabola are infinitely narrow. Roughly 2000 years before Isaac Newton and Gottfried Leibniz discovered calculus, Archimedes was using concepts of calculus. Today, similar methods of adding polygons are known as Riemann Sums.
https://segue.middlebury.edu/index.php?&site=fyse1176a-f06§ion=14424&page=63786&action=site
The anonymous private collector who bought the palimpsest for $2 million at auction in 1998 has loaned the manuscript to The Walters and is funding the studies. The studies have revealed surprising finds, including that Archimedes was the first Greek to use infinity and to set rules for infinity. Scholars have also advanced the reading and allowed the first interpretation of the Stomachion, which solely survives on one page of the palimpsest. This treatise deals with combinatorics—the number of ways a problem can be solved—which is used in modern computation.
Is this the greatest message board on the Internet or what?
I think he’d have a better chance of understanding a first- or second-generation calculus text than a modern one. To a large extent, modern calculus books are all about functions, and they rely quite heavily on algebra, which would cause him some problems, as Kimstu noted. If I understand correctly, the early developers of calculus (Newton, Leibniz, the Bernoullis, etc.) took a more geometric approach that would have been closer to that of the Greeks.
Archimedes is universally described as among history’s greatest mathematicians, but perhaps lay people (such as myself) are unduly influenced by Hollywood’s portrayal of almost effortless genius, as depicted in “A Beautiful Mind.” An intellect that makes short work of most anything it encounters has almost certainly never existed. Even Newton, Leibniz, Euler, Gauss, Pascal and other titans struggled for years to make sense of work that an above-average math grad student today would consider straightforward. I think my initial imagining was that of Archimedes finding the calculus text, nodding his head in a big “Aha!” and then turning page after page in a series of rapid-fire eureka moments. More likely, he would have spent many a week scratching his head and wishing for a companion “Intro Algebra for Dummies.”
I don’t think he’d grasp everything right away (I for damn sure didn’t w/o lots of class time), but the reason the mathematicians you mentioned had such a tough go of it was they were inventing the fields in question.
I was actually more annoyed by an episode of ST Voyager (hate the whole series, too BTW) where it was pretty much said that because they had a huge base of science, they were smarter than Leonardo Davinci (there due to some holodeck malfunction or something) despite none of the cast being able to hold a candle to him.
It would no doubt still be difficult for him, but most texts (mine at least) have useful diagrams and pictures that I think would be sufficient (given a strong mathematical and logical intellect as Archimedes had) to work out the symbols for new operations and such.
I’d wonder more if he would have taken the time. I seem to recall someone saying that there wasn’t much effort taken towards the application of ‘scientific thinking’ to any physical effect. If so, it would mean that if he didn’t find it interesting, he wouldn’t pursue it.
Isn’t mathematics supposed to be the universal language? As a brilliant mathematician, he should understand it regardless of what language it’s written in.
Yeah, I’ve heard that one before. A couple professors told me Landau’s Grundlagen der Analysis was easy to read despite the fact that it’s in German. I bought a copy, and even though I’m quite familiar with the topics it covers, I’m completely lost trying to read it.
Wouldn’t you still need to know what the unfamiliar notational symbols stood for?
Most (?) of present day mathmeticians use agreed upon symbology. This standardisation wasn’t around yet, for him. Without some “Symbology Index”, he would have to figure out what the symbols mean, using the context of the equations he seems them being used in.
One big stumbling block that only Kimstu has even hinted at: the fundamental concepts of mathematics were extremely different for the Greeks.
Arithmetic and geometry were two completely separate things. There were “numbers”, like our natural numbers, and there were “quantities” like our (positive) real numbers, but they were completely different things. Even the idea that the integers live on the real line would be new.
Now “algebra”, as we understand it and as would be written in such a text, is essentially an extension of basically arithmetic ideas. (btw: I’m ignoring all the huge difficulties in the concept of an “unknown” which didn’t come along until a few centuries ago). However, the closest ideas Archimedes would have had were geometric constructions. To get algebra and geometry together for the purposes of analysis (“calculus”) you need… wait for it… analytic geometry, which didn’t come along until Descartes.
The upshot is, even the people saying he’d have to pick it up like students do today are overstating it. Archimedes basically has geometry and trigonometry and some fairly involved arithmetic. He lacks the concept of identifying number and magnitude (“arithmetic numbers and geometric numbers are the same things”), and from there lacks what we’d include in a precalculus course, like Cartesian graphs, abstract functions, algebra 2… No, as it stood he’d be lost.
Of course, once he had those concepts he’d pick up on the ideas extremely quickly. He all but had integration down, and differentiation isn’t that much harder. The language and concepts of what we call “The Calculus” are so radically different from the way mathematics was done then that it’d be [strike]Greek[/strike] gibberish to him.