The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents and **secants** for every minute of the quadrant to ro places; there were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613).

It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents and **secants**, at intervals of a hundredth of a degree.

The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hYPerbolic log sines, tangents and **secants** for every minute of the quadrant to 5 places of decimals.

With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and **secants**.'

And from this, and from the property of a rigid body, already stated in 29, it follows, that the components along a is ne of connection of all the points traversed by that line, whether -in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connection are to each other inversely as the cosines, or directly as the **secants**, of the angles made by the paths of those points with the line of connection.