Logarithms discovered by Napier in 1614 were based on sine tables with 0.9999999 just below sine 90 degrees as the base which is raised to successive powers. Initially the results are nearly equal to the shortfall from 1.0000000. It would be a very onerous task to raise these powers from sine 90 degrees down to sine 1 degree, but this would be helped by by sine 75 degrees equalling 0.9659258 being raised to the power of 10 and equalling sine 45 degrees which is 0.7070299. Without these tables of logarithms there would be no theory from Nicholas Mercator of the area under a symmetrical hyperbola equalling the log of the distance along the x axis, nor of Isaac Newton's reversion of the hyperbola formula to achieve the infinite series for the antilogarithm e. This year is the 400th anniversary of Napier's discovery which is not being properly commemorated largely because modern mathematicians have no idea how Napier achieved it. submitted by Peter L. Griffiths.

## Napier's logarithms

Logarithms discovered by Napier in 1614 were based on sine tables with 0.9999999 just below sine 90 degrees as the base which is raised to successive powers. Initially the results are nearly equal to the shortfall from 1.0000000. It would be a very onerous task to raise these powers from sine 90 degrees down to sine 1 degree, but this would be helped by by sine 75 degrees equalling 0.9659258 being raised to the power of 10 and equalling sine 45 degrees which is 0.7070299. Without these tables of logarithms there would be no theory from Nicholas Mercator of the area under a symmetrical hyperbola equalling the log of the distance along the x axis, nor of Isaac Newton's reversion of the hyperbola formula to achieve the infinite series for the antilogarithm e. This year is the 400th anniversary of Napier's discovery which is not being properly commemorated largely because modern mathematicians have no idea how Napier achieved it. submitted by Peter L. Griffiths.