at any assigned height in the Air; and è contra, having the height of the Mercury given, to find the height of the Place where the Barometer stands, are Problems not more difficult than curious; and which I thus resolve.
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 ,