The Expansions of the Air being reciprocally as the heights of the Mercury, it is evident, that by the help of the Curve of the _Hyperbola_ and its _Asymptotes_, the said Expansions may be expounded to any given height of the _Mercury_: For by the 65th _Prop. lib. 2. Conic. Mydorgii_, _the Rectangles_, _ABCE_, _AKGE_, _ALDE_, _&c._ (in _Plate 1. Fig. 4._) are always equal, and consequently the sides, _CB_, _GK_, _LD_, &c. are reciprocally as the sides _AB_, _AK_, _AL_, &c. If then the Lines _AB_, _AK_, _AL_, be supposed equal to the heights of the _Mercury_, or the pressures of the Atmosphere, the Lines _CB_, _GK_, _LD_, answering thereto, will be as the Expansions of the Air under those Pressures, or the Bulks that the same quantity of Air will occupy; which Expansions being taken infinitely many, and infinitely little, (according to the Method of Indivisibles) their Summ will give the Spaces of Air between the several heights of the _Barometer_; that is to say, the Summ of all the Lines between _CB_ and _KG_, or the _Area_ _CBKG_, will be proportioned to the Distance or Space intercepted between the Levels of two Places in the Air, where the _Mercury_ would stand at the heights represented by the Lines _AB_, _AK_; so then the Spaces of Air answering to equal Parts of _Mercury_ in the _Barometer_, are as the _Area's_ _CBKG_, _GKLD_, _DLFM_, &c. These _Area's_ again are, by the Demonstration of _Gregory_ of St. _Vincent_, proportionate to the _Logarithms_ of the Numbers expressing the _Rationes_ of _AK_ to _AB_, of _AL_ to _AK_, of _AM_ to _AL_, &c. So then by the common Table of _Logarithms_, the height of any Place in the _Atmosphere_, having any assign'd height of the _Mercury_, may most easily be found: For the Line _CB_ in the _Hyperbola_, whereof the _Area's_ design the _Tabular Logarithms_, being 0,0144765; 'twill be, as 0,0144765, to the difference of the _Logarithms_ of 30, or any other lesser Number, for 900 Feet, or the Space answering to an Inch of _Mercury_, if the Air were equally prest with 30 Inches of _Mercury_, and every where alike, to the height of the _Barometer_ in the Air, where it will stand at that lesser number of Inches: And by the Converse of this Proportion may the height of the _Mercury_ be found, having the Altitude of the Place given. From these Rules I deriv'd the following Tables. _A Table shewing the Altitude, to given heights of the _Mercury_._ Inch. Feet. 30 0 29 915 28 1862 27 2844 26 3863 25 4922 20 10947 15 18715 10 29662 5 48378 1 91831 0.5 110547 0.25 129262 0.1 29 mil. or 154000 0.01 41 m. or 216169 0.001 53 m. 278338 _A Table shewing the heights of the _Mercury_, at given Altitudes._ Feet. Inch.