This paper presents a new proof of Archimedes' volume propositions for conoids and spheroids that unifies the treatment of all three conic sections. While Archimedes established that solids generated by rotating conic sections maintain specific volume ratios to cones with the same base and axis, his original arguments-both the exhaustion method in On Conoids and Spheroids and the mechanical approach in Method-require distinct arguments for each type of conic section. Our approach transforms both the conic solid and its reference cone into combinations of cone-related figures through an axis-stretching technique, then establishes volume ratios using only elementary proportional relationships. This unified geometric treatment reveals the underlying structural unity between different conic sections, while employing only tools from Euclid's Elements and techniques found throughout Archimedes' works.
We combine ideas from geometry, calculus, complex numbers, and linear algebra to find a simple way to characterize the tendency of solutions of a 2 & times;2 linear system of differential equations to rotate in an averaged sense. This characterization extends to solutions that do not truly rotate by giving geometric information about the eigenvectors or generalized eigenvectors of the corresponding coefficient matrix. Toward these ends, we introduce a scalar quantity that we name the rotational trace. We show that the sign of the rotational trace predicts the direction of rotational motion in an averaged sense. In addition, the rotational trace is related to well-known invariants of 2 & times;2 matrices such as the trace under rotational transformations. Finally, we connect the rotational trace to vector calculus by finding finite approximations that compute the flux and rotation of the system of differential equations exactly while using as few as three points and provide a simple proof of the right-hand rule for cross-products.
We present a modern and transparent proof of the Banach-Zarecki theorem, characterizing absolute continuity via continuity, bounded variation, and the preservation of Lebesgue null sets, while avoiding cumbersome estimates and classical summation arguments.
This paper derives the maximum variance of a finite sequence of real numbers, given their mean, minimum and maximum. An example is provided in which the maximum variance is less than half of the Bhatia-Davis upper bound, (maximum - mean)(mean - minimum). As the length of the sequence increases, the maximum variance under these constraints approaches this bound.
