By G. H. Hardy

There will be few textbooks of arithmetic as famous as Hardy's natural arithmetic. on the grounds that its e-book in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become firstly 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 basic principles of the differential and imperative calculus, of the houses of limitless sequence and of alternative subject matters related to the suggestion of restrict.

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Latin text and translation [160] IV The angles between the generator and the directrix will be called the angles with the directrix. V The fixed point about which the moving angle rotates will be called the pole. VI The part of the generator cut off between the pole and the directrix [104] will be called the interval. VII The arm of the moving angle that is carried along by the generator will be called the passive arm. VIII The other arm, which is intersected by the generator, will be called the active arm and, after extension beyond the vertex of the angle, will be called the active line.

Theorem XIV. A chord connecting two points on an ellipse and bisected by a diameter either passes through the center or is ordinate-wise applied to this diameter. COROLLARY 1. A diameter in an ellipse that bisects a chord (not passing through the center) will also bisect all chords parallel to that chord. COROLLARY 2. A straight line in an ellipse that bisects two parallel chords passes through the center and is therefore a diameter. COROLLARY 3. A chord connecting two arbitrary points on an ellipse lies entirely inside the ellipse.

B H crlls patiens. B G crNS efficiens. P G linea efficiens. D K deJcri6ens in ftatjone prima, five aefcri/mJJ limpliciter. D Be, DBA anguli mobiles in patione prima. ttiolte prima, live eJJiciens fimpliciter. ue a"gulos mohiles D Be, DBA. THEOREMA Propojitio I. I. lt qurevis recta a quolibct curvre puncro ad ~eflribelltemcffi;' ciellti :equidifians applicata poffit recrangulum, fub in. lure intcr PO/11m& applicatam intercipitur, contentum. ~l:1libet efficiellte, Sit efficiente ABC ,. '-'ltervallo B D, & direElrice E F dcrcripta ita tIt angfllfls mobi/isD B A fit requalis angulo E D B, ad direClricem ,.