This chapter discusses some of the recalls the sophisticated use of logarithms to measure and compare ratios found in the musical work of pietro Mengoli. The use of logarithms was capable of solving the problem of comparing musical intervals, without resorting to approximations or redefinitions of the intervals. Similarly, the subtraction of logarithms corresponds to the subtraction of musical intervals, and the multiplication of a logarithm by a number corresponds to the multiplication of a musical interval by the same number. The first person to use logarithms to produce a diagram of musical pitch, and hence a wholly new kind of quantification of pitch, was Rene Descartes: Nicolaus Mercator and Isaac newton both followed him, but none of the three published the details of this mathematical work. Issac Newton's writings on music were much more compact than those of Mercator.