By George Boole
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Extra info for A Treatise on Differential Equations
Kempner, who in 1914 considered what would happen if all terms are removed from it which have a particular digit appearing in their denominators. For example, if we choose the digit 7, we would exclude the terms with denominators such as 7, 27, 173,33779, etc. There are 10 such series, each resulting from the removal of one of the digits 0, 1, 2, ... , 9, and the first question which naturally arises is just what percentage of the terms of the series are we removing by the process? For example, if we remove all terms involving 0 we are left with 1 1 1 1 1 1 1 + - + - + ...
By their assistance the Mariner steers his Vessel, the Geometrician investigates the Nature of higher Curves, the Astronomer determines the Places of the Stars, the Philosopher accounts for other Phenomena of Nature; and lastly, the Usurer computes the Interest of his Money. The work found the particular affections of Henry Briggs (1561- I630), who had become the first Professor of Geometry at Gresham College, London, in 1596; in 1620 he was to become the first occupant of the Savilian Chair of Geometry in Oxford; later we will meet the great G.
They use the now standard labelling for a triangle (spherical or plane) that capital letters represent the vertices and the corresponding small letters the side opposite (as we mentioned in the introduction, yet another inheritance from Euler) and it should be borne in mind that any side of a spherical triangle can be thought of as the angle it subtends at the centre of the defining circle. 7. 7. Napier's circle. Napier's rules are, then, • the sine of any point equals the product of the tangents of the adjacent points, • the sine of any point equals the product of the cosines of the opposite points.