The computation of the volumes of spheres can be done without actually doing any integrals. We only need to know the recusion that B(n) = S(n-1)/n and S(n)=2π B(n-1) and the induction assumption S(0)=2 and B(0)=1. The first recursion follows from integrating the sphere volume up (make shells). The second is an Archimedes trick: the cylinder surface area enclosing a sphere is the same than the surface area because there is an area preserving map from the sphere to the cylinder (project from the center ball which is in the case n=2 the 1-ball (the z axes in the case of Archimedes). Remark:: My video from February 2, 2026 states the Archimedes trick |Sⁿ| = |S¹| |B^n-1| in higher dimensions. It can also be found in this 3Blue1Brown video from February 27'th 2026. This got me interested in the history of the Archimedean property in higher dimensions. I had thought that this pedagogical point had been original in my teaching as I had never thought about the higher dimensional case before 2018 in this lecture. from 2018, where I had used the argument already. A conversation with ChatGPT 5.2 done on February 28th 2026 got me the reference The Archimedean Projection Property by Vincent Coll, Jeff Dodd and Michael Harrison from 2015. As for now, this is the earliest reference pointing out the simple idea that Archimedes projection also works in higher dimensions. It is earlier than 2018. I have not seen the argument yet in textbooks but would not be surprised if it would. The article of Coll Dodd and Harrison say it is "well known" (I think that "well known" would deserve that least one reference to a textbook, even so it is obvious once seen; most things taught in undergraduate courses is "obvious". There still can be innovation in presentation). There can still also be innovation in presenting the obvious in a new way. Here is the chat (reformatted to HTML and shortened a bit and changed some notation like φ instead of π used by the bot, since π was used as the number π).

Me	Chat GPT 5.2
"Give me a source for the higher dimensional statement that the volume of the n-dimensional sphere is 2Pi times the volume of the (n-1)- dimensional ball. In dimension 2 this means that the volume of the 2 dimensional sphere is 2pi times the volume of the 1 dimensional ball, a fact attributed to Archimedes. I would like to know who first pointed out the higher dimensional generalization of this fact."	Greg Huber's short note in "American Mathematical Monthly" derives the sphere/same gamma-function formulas from Gaussian integrals. Who first pointed out the higher-dimensional generalization? The honest answer is: I can't find a definitive primary-source sentence like |Sn| = 2 π |B(n-1)| attributed to a specific 19th-century author in the open sources I can access quickly. What is pretty clear from the history-of-math literature is: systematic work on n-dimensional spherical geometry / volumes is commonly traced to Ludwig Schlaefli, whose treatise was written in the early 1850s (published later), with an English excerpt/notice by Arthur Cayley (1858). By the time these general n-dimensional volume formulas were circulating, the identity you want is an immediate corollary of the closed forms. These closed forms and the recurrence are stated together in modern references. So, if what you need is a historically plausible first, the safest statement I can support with sources is: Schlaefli (early 1850s; via Cayley 1858 and later expositions) is one of the earliest widely cited origins for the general n-dimensional sphere/ball volume formulas, and the relation is a straightforward consequence of those formulas.
"I'm interested who first pointed out that in higher dimensions, the analog of the Archimedes argument can be used. I pointed this out in my video and had thought that the higher dimensional Archimedes projection analogy was original (like here).	Archimedes original work "On the Sphere and Cylinder" gives the 3-dimensional volume of a sphere by comparing cross-sections with those of a cylinder and cone and using what we now call Cavalieri's principle or the method of indivisibles, though Archimedes himself did not phrase it that way. Archimedes didn't state anything about higher dimensions; his argument was purely for the ordinary 3-sphere/ball and circle/area case. Archimedes provided the original 3-dimensional argument, but he did not make a higher-dimensional analog. Cavalieri's principle (17th century) is the first systematic formulation of the slicing method you are using, and it implicitly generalizes to higher dimensions, even if Cavalieri himself didn't state the precise formula you used. Modern texts (20'th century) articulate the recurrence and explicitly note that slicing/Cavalieri arguments generalize to arbitrary dimension. So, your observation, while elegant and maybe novel in the explicitly stated form you gave it is actually a natural extension of Cavalieri's slicing method as used in integral geometry, and that is the first identifiable historical source for the underlying idea that makes your analog work.
"I'm interested in the projection argument which shows that the area of the 2-sphere is 2 π times the volume of the 1 ball: 4 π=2π*2. This identity shows that the area of the 2-sphere is equal to the area of the cylinder enclosing the sphere. One can use the same argument in higher dimensions. Project the n-sphere onto the product of the 1-sphere with the (n-1)-ball. This projection preserves volume. This then proves that |Sn| = 2π |B(n-1)|. Can you point me to a reference, where this statement is proven using the volume preserving map from the sphere to the cylinder in higher dimensions?"	Yes, what you're describing a volu.me/area-preserving Archimedean map from the sphere to a cylinder S1 x B(n-1) giving Sn is essentially the co-dimension 2 Archimedean projection property. The cleanest reference that states (and uses) exactly this measure-preserving idea in all dimensions is Vincent Coll, Jeff Dodd, Michael Harrison, "The Archimedean Projection Property". Early in the paper they explicitly state the higher-dimensional generalization: Let Sn-1 in Rn and φ be the codimension-2 orthogonal projection restricted to the sphere, so φ(Sn-1) = B(n-1) Then for every measurable U in B(n-2), one has |φ-1 U| = 2 π |U|. For the classical area-preserving statement (framed in differential forms/symplectic language), see Nicholas Proudfoo's 2011 notes which explicitly say Archimedes observed the lateral projection from the cylinder to to the sphere is area preserving.