1. 1. It may not be generally known how much more rapidly such operations as multiplying or dividing by π, or reducing metres to yards, etc. can be effected by the methods of “Practice” than by direct multiplication and division.

2. 2. For example, the commonly used approximation for the value of π, though an excellent first approximation, is only true to three significant figures; but (i) is true to five significant figures, and thus yields a second approximation by a correction of the first. The value (ii) which is slightly higher, may be used as a useful check.

   

Some idea of a system of such equations may be given in order to clear the way.

In Professor Chrystal’s Algebra, Chapter XIV., No. 18, p. 303, we have:

“Of the equations

with two similar equations in y, z, where x, y, z are unequal; prove that any two imply the third.” (Cayley.)

This begins with an “Epistle Dedicatory” by Edm. Wingate and the Preface of John Kersey. The Table of Contents divides the whole book into three parts; one half of the first portion was Wingate’s and the other half was Kersey’s improvement upon Wingate, while nearly the whole of the second part and all the Appendix are comprised in “chapters that have been entirely composed by the said J. K.”

After Pythagoras and the Italic school, Professor Loria takes up the Eleatics, the Atomists, the Sophists. He expounds the arguments of Zeno (born about the beginning of the 5th century B.C.) against multiplicity and motion, and attributes to the Eleatic philosopher—he was not strictly a mathematician—the great merit of having for the first time discussed infinitesimal and infinite quantities as well as infinite numbers.