MATHEMATICAL ESSAYS AND RECREATIONS

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Side 41 - But hail, thou goddess sage and holy! Hail, divinest Melancholy! Whose saintly visage is too bright To hit the sense of human sight, And therefore to our weaker view O'erlaid with black, staid Wisdom's hue; Black, but such as in esteem Prince Memnon's...
Side 79 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
Side 27 - The intrinsic character of mathematical research and knowledge is based essentially on three properties: first, on its conservative attitude towards the old truths and discoveries of mathematics; secondly, on its progressive mode of development, due to the incessant acquisition of new knowledge on the basis of the old; and thirdly, on its self-sufficiency and its consequent absolute independence. — SCHUBERT, H. Mathematical Essays and Recreations (Chicago, 1898), p.
Side 127 - ... that he must have exercised in calculating the limits of n without the advantages of the Arabian system of numerals and of the decimal notation. For it must be considered that at many stages of the computation what we call the extraction of roots was necessary, and that Archimedes could only by extremely tedious calculations obtain ratios that expressed approximately the roots of given numbers and fractions. With regard to the mathematicians of Greece that follow ArchiThe later mathema- niedes,...
Side 116 - ... means the case. On the contrary, for some two hundred years, the endeavors of many considerable mathematicians have been solely directed towards demonstrating with exactness that the problem is insolvable. It is, as a rule, — and naturally, — more difficult to prove that something is impossible than to prove that it is possible. And thus it has happened, that up to within a few years ago, despite the employment of the most varied and the most comprehensive methods of modern mathematics, no...
Side 142 - Since the beginning of this century, consequently, the efforts of a number of mathematicians have been bent upon proving that n generally is not algebraical, that is, that it can not be the root of any equation having whole numbers for coefficients. But mathematics had to make tremendous strides forward before the means were at hand to accomplish this demonstration. After the French academician, Professor Hermite, had furnished important preparatory assistance in his treatise Sur la Fonction Exponentielle...
Side 28 - Part 3, chap. 1, sect. 8. 243. Whenever ... a controversy arises in mathematics, the issue is not whether a thing is true or not, but whether the proof might not be conducted more simply in some other way, or whether the proposition demonstrated is sufficiently important for the advancement of the science as to deserve especial enunciation and emphasis, or finally, whether the proposition is not a special case of some other and more general truth which is as easily discovered. — SCHUBERT, H. Mathematical...
Side 122 - The rule given in this papyrus for obtaining a square equal to a circle, specifies that the diameter of the circle shall be shortened oneninth of its length and upon the shortened line thus obtained a square erected. Of course, the area of a square of this construction is only approximately equal to the area of the circle. An idea may be obtained of the degree of exactness of this original, primitive quadrature by our remarking...
Side 141 - Zurich employd some dec- clirious metho<lades ago to compute the value of n to 3 places. The floor of a room is divided up into equal squares, so as to resemble a huge chess-board, and a needle exactly equal in length to the side of each of these squares, is cast haphazard upon the floor. If we calculate, now, the probabilities of the needle so falling as to lie wholly within one of the squares, that is so that it does not cross any of the parallel lines forming the squares, the result of the calculation...
Side 126 - ... and more famous, from a new point of view. Hippocrates was not satisfied with approximate equalities, and searched for curvilinearly bounded plane figures which should be mathematically equal to a rectilinearly bounded figure, and therefore could be converted by ruler and compasses into a square equal in area. First, Hippocrates found that the crescent-shaped plane figure produced by drawing two perpendicular radii in a circle and describing upon the line joining their extremities a semicircle,...

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