We present a short proof of Descartes Circle Theorem on the “curvaturecenters” of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We also prove an analogue for spheres. It can be traced back to at least Descartes that four mutually tangent circles have curvatures (reciprocals of radii) satisfying the relatio...

In this work, we formally proved Descartes Rule of Signs, which relates the number of positive real roots of a polynomial with the number of sign changes in its coefficient list. Our proof follows the simple inductive proof given by Arthan [1], which was also used by John Harrison in his HOL Light formalisation. We proved most of the lemmas for arbitrary linearly-ordered integrity domains (e.g....

JOSEPH MEITES, BERNARD T. DONOVAN, and SAMUEL M. McCANN (editors), Pioneers in neuroendocrinology, volume II, New York and London, Plenum Press, 1978, 8vo, pp. 422, $39.00. This series of autobiographies continues with twenty-four further essays. The individuals concerned include the three editors, Roger Guillemin, and Solly Zuckerman. These are the second generation of pioneers, whose contribu...

OBJECTIVE The problem of the substantial union of the soul and the body and therefore the mechanisms of interaction between them represents the core of the Cartesian dualistic philosophy. This philosophy is based upon a neuroanatomical obvious misconception, consisting mainly on a wrong intraventricular position of the pineal gland and its capacity of movement to act as a valve regulating the f...