Quad (figure skating)
A quad, or quadruple, is a figure skating jump with at least three but less than four revolutions. Most quadruple jumps have 3 1/2 revolutions; the quadruple Axel has 4 1/2 revolutions, although no figure skater to date has completed this jump in competition. The quadruple toe loop and quadruple Salchow are the two most commonly skated in the discipline. Quadruple jumps have become increasingly common among World and Olympic level men's single skaters, to the point that not having one in a program has come to be seen as a severe handicap. The first person to land a ratified quadruple jump in competition was Canadian Kurt Browning in 1988. In 2002, Miki Ando became the first woman to do so, and she remains one of only two women to have landed a ratified quadruple jump in competition.
Pappus of Alexandria
Of Book IV the title and preface have been lost, so that the program has to be gathered from the book itself. At the beginning is the well-known generalization of Euclid I.47 (Pappus's area theorem), then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two. This and several other propositions on contact, e.g. cases of circles touching one another and inscribed in the figure made of three semicircles and known as arbelos ("shoemakers knife") form the first division of the book; Pappus turns then to a consideration of certain properties of Archimedes's spiral, the conchoid of Nicomedes (already mentioned in Book I as supplying a method of doubling the cube), and the curve discovered most probably by Hippias of Elis about 420 BC, and known by the name, τετραγωνισμός, or quadratrix. Proposition 30 describes the construction of a curve of double curvature called by Pappus the helix on a sphere; it is described by a point moving uniformly along the arc of a great circle, which itself turns about its diameter uniformly, the point describing a quadrant and the great circle a complete revolution in the same time. The area of the surface included between this curve and its base is found – the first known instance of a quadrature of a curved surface. The rest of the book treats of the trisection of an angle, and the solution of more general problems of the same kind by means of the quadratrix and spiral. In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the focus and directrix.