"Squaring the Circle": A History of the Problem |
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Common terms and phrases
algebraic equation algebraic numbers already determined analytical angles Arithmetic axiom of Archimedes B₁ B₂ calculated century chord circle squarer circumference circumscribed coefficients are rational continued fractions convergent coordinates corresponding Crelle's Journal curve denote diameter different from zero E. W. HOBSON employed equal equation with rational equivalent established Euclidean construction Euclidean determination Euclidean Geometry Euclidean problem Euler expression finite number formula given Greek Mathematicians hexagon Huyghens ideal problem inscribed polygon integer investigation limits logarithms lunulae magnitude Mathematicians means method of Archimedes method of exhaustions mode notation obtained perimeter places of decimals postulations problem of squaring proof proved quadrable quadrature radius rational coefficients rational functions rational number rectification regarded regular polygon relation result segment semi-circle sequence shewed shewn sides solution square roots squaring the circle straight line symmetrical function tan-¹ tangent theorem transcendental number triangle Trigonometry Wallis x₁
Popular passages
Page 13 - Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.
Page 16 - Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be hft some magnitude which will be less than the lesser magnitude set out.
Page 13 - Or, to take a case yet stronger, when it is affirmed, that " the area of a circle is equal to that of a triangle having the circumference for its base, and the radius for its altitude...
Page 17 - Euclid xn. 2, that the areas of two circles are to one another as the squares on their diameters.
Page 41 - ... surgeon, we elect legislators to operate on the body politic without any scientific training, and without any adequate knowledge of social science, or even of the principles of legislation. It is therefore very fortunate that the legislators I have referred to, or indeed any others, could not alter the ratio of the circumference to the diameter of a circle by the one billionth part of a unit, Even in Canada we now talk in billions, and shall soon have to learn in the school of experience to pay...
Page 19 - Proposition 3. The ratio of the circumference of any circle to its diameter is less than 3| but greater than...
Page 19 - PROPOSITION 1 The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle. Let ABCD be the given circle, K the triangle described.
Page 4 - The Academy has resolved, this year, to examine no longer any solutions to problems on the following subjects: The duplication of the cube, the trisection of the angle, the quadrature of the circle, or any machine claiming to be a perpetuum mobile.
Page 16 - ... From the time of Plato (429—348 BC), who emphasized the distinction between Geometry which deals with incorporeal things or images of pure thought and Mechanics which is concerned with things in the external world, the idea became prevalent that...