Each infinitesimal PR joins to its neighboring infinitesimals PR in a manner characteristic of a cone. For an infinitesimal not only possesses a certain extent (that is to say, a certain length we can speak meaningfully of PR’s length in one direction even if it is infinitely thin in another direction), but it also abuts its neighboring infinitesimals in some characteristic way. I don’t think a cone’s infinitesimal and a sphere’s infinitesimal are quite the same thing. Not quite equal, I think all the infinitesimals PR are adding up to equal a cone, while the KL infinitesimals are adding up to create a sphere. Surely, at this location, the infinitesimals of the cone and the sphere are equal, right?
This is an intersection of the cone AEF and the sphere. Notice the intersection that I have circled. Now, as mathematical objects, infinitesimals possess a certain slippery property that I want to explore. Of course, those infinitely tiny pieces are called infinitesimals. Their grand sum works out to be a finite quantity. That’s all it is infinitely tiny pieces added together infinitely many times. In this case, Archimedes has managed to construct from infinitesimals both a cone and a sphere.Īnd that is an integral. Infinitely many of the infinitely small adds up to something. Any sum of infinitely thin things would be zero, except that in this case we have infinitely many of them. There are infinitely many circles of diameter PR (a bit of a misnomer there there are infinitely many diameters, ranging in length from 0 to EF, all of which we are calling PR), all infinitely thin. The document is in fragments, but here is one unbroken chunk of it:įinally, Archimedes had previously established that AQ was equal to PQ. In the twentieth century, we found a lost work of his called The Method, and it details numerous theorems that depend on both the integral and on his theory of the lever. But all of these ideas employ infinitesimals, an idea that most classical mathematicians, Archimedes included, found abhorrent.
#Does calculus need infinitesimals plus
Had Archimedes also discovered derivatives, plus the fact that the integral of a function equals its anti-derivative, then modern science might have developed some 2,000 years earlier than it did. Integrals were known to Archimedes, arguably the greatest mathematician humanity ever produced. I think it is easiest if we start with integrals. An infinitesimal is a strange mathematical object, but once we get a good understanding of what they are, then integrals, differentials, and the Fundamental Theorem of Calculus all drop into place. Calculus started making a lot more sense for me once I started looking at it in terms of infinitesimals.
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