The butterfly theorem is proved by assigning point masses to the four vertices of the wings and using the distributive property of the mass centre of a mechanical system.

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year…

A physical pendulum with a magnetic load is proposed for comparison of linear and nonlinear oscillations. The magnetic load is repelled by permanent magnets which are disposed symmetrically relative to the load. It is established that positions of the pendulum and the magnets determine the dependence of restoring force on displacement of the load.…

This paper deals with a brief history of the most remarkable Euler numbers "e,"?"i"?and?"?" in mathematical sciences. Included are many properties of the constants "e,"?"i"?and?"?" and their applications in algebra, geometry, physics, chemistry, ecology, business and industry. Special…

Archimedes calculated the centre of gravity of the cone but the proof of this theorem is not extant in his works. Knorr made a reconstruction of this proof utilizing geometrical arguments. This paper proves this theorem by means of a physical demonstration utilizing the law of the lever, and by adapting from Archimedes the method of mechanical…

The circle of Apollonius is named after the ancient geometrician Apollonius of Perga. This beautiful geometric construct can be helpful when solving some general problems of geometry and mathematical physics, optics, and electricity. Here we discuss two of its applications: localizing an object in space and calculating electric fields. First, we…

The free-energy change, or binding energy, of an idealized bubble cluster is calculated on the basis of one mole of gas, and on the basis of a single bubble going from sphere to polyhedron. Some new relations of bubble geometry are developed in the course of the calculation. (BB)

Mathematical theorems about the properties of ellipsoids are developed. Included are Poisson's theorem concerning the magnetization of a homogeneous body of ellipsoidal shape, the polarization of a dielectric, the transport of heat or electricity through an ellipsoid, and other problems. (BB)

Before the 19th century the idea of more than three dimensions was exceptional. During the 19th century, however, geometry was revolutionized and new branches were developed. This revolution also created the idea of the possibility of a n-dimensional geometry or space; flatland, i.e. n = 2, was a consequence of this new thinking. In 1884 the…

Examples are given of proofs to standard geometric theorems using physics and mechanics. (PK)

The second of three volumes of a mathematics training course for Navy personnel, this document contains material primarily found at the college level. Beginning with logarithms and trigonometry, the text moves into vectors and static equilibrium (physics). Coordinate geometry, conic sections, and the tangents, normals, and slopes of curves follow.…

A simple geometric construction giving acceptable approximations to the locations of twelve points for placing frets on a guitar is described both geometrically and analytically. (DT)

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

The Poincare group, the group of transformations of the plane which preserve the Minkowski distance between points, is derived as compositions of suitably defined reflections in straight lines. It is shown that any such transformations must be one of four types. (Author/JN)

At the pre-college and first-year college level of physics instruction, Kepler's laws are generally taught as empirical laws of nature. Introductory physics textbooks only derive Kepler's Second law of areas. It is possible to derive all of Kepler's laws mathematically from the conservation laws, employing only high-school algebra and geometry.…