Archimedes gave geometrical demonstrations to find the volume of a sphere and the ellipsoids of revolution in totally different ways, although both lead to the same integral,∫[2a,0]x (2a — x)dx. Nicolas Bourbaki -a group of French mathematicians-state their views on why Archimedes had no concept of integral calculus as follows: "Might it not be that Archimedes regarded such a standpoint as extreme 'abstraction,'and dared to concentrate on studying characteristic properties of each figure he was working on ?" Certainly there is something in what Nicolas Bourbaki say. However, their views do not answer fully the question To the solution of this difficult problem, in my opinion, an important clue can be found by considering Archimedes' scholastic career in chronological order. It was not until Archimedes wrote On Spirals in his late forties or early fifties that he could work out the summing of the series 1²+2²+…n². In his later work On Conoids and Spheroids Archimedes could obtain for the first time the sum of a series ∑Xk(2a—Xk), necessary to give geometrical proofs about the volume of the ellipsoids of revolution However,it was difficult for Archimedes,in writing On the Sphere and Cylinder I, to obtain the sum. Therefore, he proved the theorem about the volume of a sphere in a way not making use of such summation When the same integral appeared, Archimedes could not notice the internal connection unifying them. This may be because, for one thing, he excluded from geometry, due to their mechanical nature, the discussions using indivisibles found in The Method which could have been a clue toward noticing the internal connection. Secondly, obtaining the sum of a series was not a simple matter to Archimedes who lacked the necessary algebraic symbols.
It has been asserted that the investigation on late radiation effects by the Atomic Bomb Casualty Commission-(ABCC)and the Radiation Effects Research Foundation (RERF) is the one and only exhaustive study for more than thirty years among the one hundred thousand atomic bomb survivors.
However, their studies had some principal problems which lead inevitably serious underestimations of radiation effects on human body. On the starting point of the research, ABCC made an exception cf the period from December 1945 till September 1950, concealing the fact that the death rate especially among the high level radiation survivors was extremely high at the period. ABCC also excluded such survivors from the research as those who resided out of the cities of Hiroshima and Nagasaki in October, 1950 because of the time lag of the reconstructions of their houses around ground zero. Morever, ABCC cut out the most of young survivors who migrated out of the cities before 1950. Their late cancer deaths should surely have raised s rate among the survivors, if ABCC investigated them.
The cancer risk of radiation exposure was estimated among these biased atomic bomb survivors by ABCC, and was substantially underestimated.
Its risk factor should be thoroughly reevaluated from the point of reestimation not only of the atomic bomb radiation doses, but of the fundamental date obtained by ABCC and RERF.
We first consider the difference of numerical solutions for algebraic equations given by Vieta and Raphson. Second, we discuss the differences between the methods to solve the algebraic equations, one is pro- posed by Newton and another is proposed by Raphson. Considering the two facts, we make an attempt to evaluate the Raphson's achievements in this field.
In his early works, W. R. Hamilton set out to describe optical and mechanical systems by a single characteristic function. It is generally accepted that he based his theory on variational principles.
In this note the author shows that Hamilton derived the characteristic function V from the law of refraction and reflection in optics, and the equations of motion in mechanics, but not from variational principles. And she also shows that it was crucial for his description of optical-mechanical systems that differentiations of both characteristic functions have the same mathematical form； i.e. exact differential 1-form.