By G. H. Hardy
There might be few textbooks of arithmetic as famous as Hardy's natural arithmetic. when you consider that its booklet in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have became at the start in their undergraduate classes. In its pages, Hardy combines the keenness of a missionary with the rigor of a purist in his exposition of the elemental rules of the differential and vital calculus, of the houses of endless sequence and of alternative themes concerning the idea of restrict.
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9, 5. A similar proof may be applied to show that when of a quadrilateral are given the area a is the four sides maximum when CO. A 0=BO. , when the figure is cyclic. Problem Papers, 1888, 7. To draw a See Milne's Companion to the Weekly p. ] parallel to a given line meeting a semicircle in D such that A BCD is a quadrilateral of maximum area. [As before, when A BCD is a maximum it is equal C and to the consecutive area ABC'D'. J*=c 2/c*and Making C and D on the diameter have these substitutions in (1) a.
In the particular case when a = b - c, the equation for d reduces to d = 2a, hence thus showing that the quadrilateral is half the regular inscribed hexagon. MAXIMUM AND MINIMUM. } The diagonals of a quadrilateral are 9 and 10 feet and two opposite sides 5 and 3 feet find when its area is a maximum. 6. ; 10. and Theorem. Having given the locus of the vertex a the base line L A B of a triangle meeting the base pro- AC+BO is a minimum duced, the sum of the sides L is the external bisector of the vertical angle.
If points P, Q, and R be taken on the sides of a triangle the circles AQR, BRP, and GPQ pass through a common point 0. For let the circles AQR and BRP meet in 0. Then since (Euc. III. 22) QOR = 7r-A and jROP = 7r-JS, we have QOP = 27r-(7r-^)-(7r- B) = ^+ = 7r-C; there19. J fore the quadrilateral Tfie angles POQO is cyclic. BOG, CO A, AOB, subtended by the given triangle at 0, are respectively is within the triangle ABC. when the sides of A+P, B+Q, C+R, it For, applying Euc. follows that I. 32 to the triangles jBOCand LAOB=C+CAO + CBO.
A course of pure mathematics by G. H. Hardy