VOL. XVI.I PHILOSOPHICAL TRANSACTIONS. 301 



but because it was summer weather, and consequently the air rarefied, when 

 these experiments were made, we may, without sensible error, say in round 

 numbers, that the barometer standing at 30 inches, and in a mean state of heat 

 and cold, the specific gravity of the air to water is as 1 to 800. By the like 

 trials, the weight of mercury to water is as 13-J- to J, or very near it ; so that 

 the weight of mercury to air is as 10,800 to 1, and a cylinder of air of 10,80O 

 inches or QOO feet is equal to an inch of mercury ; and were the air of an equal 

 density like water, the whole atmosphere would be no more than 5,1 miles 

 high ; and in the ascent of every QOO feet the barometer would sink an inch. 

 But the expansion of the air increasing in the same proportion as the incumbent 

 weight of the atmosphere decreases, that is, as the mercury in the barometer 

 sinks, the upper parts of the air are much more rarefied than the lower, and 

 each space, answering to an inch of quicksilver, grows still larger ; so that the 

 atmosphere must be extended to a much greater height. Now on these prin- 

 ciples, to determine the height of the mercury at any assigned height in the 

 air, and e 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 cu- 

 rious, and which I thus resolve. 



The expansions of the air being reciprocally as the heights of the mercury, 

 it is ^ident, that by means 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 A B C E, A K G E, 

 AL D E, &c. (in fig. T , pi. Q) are always equal, and consequently the sides C B, 

 G K, L D, &c. are reciprocally as the sides, A B, A K, A L, &c. If then the 

 lines A B, A K, A L, be supposed equal to the heights of the mercury, or the 

 pressures of the atmosphere, the lines C B, KG, L D, answeriqg thereto, will 

 be as the expansions of the air under those pressures, or the spaces that the 

 same quantity of air will occupy ; which expansions being taken infinitely many 

 and infinitely small, according to the method of indivisibles, their sum will 

 give the spaces of air between the several heights of the barometer ; that is 

 the sum of all the lines between C B and K G, or the area C B K G, will be pro- 

 portional 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 A B, A K ; so then the spaces of air answering to equal parts of mercury 

 in the barometer are as the areas CBKG, GKLD, DLFM, &c. These areas 

 again are, by the demonstration of Gregory of St. Vincent, proportional to the 

 logarithms of the numbers expressing the ratios of A K to A B, of A L to A K, 

 of A M to A L, &c. So then by the common tables of logarithms, the height 

 of any place in the atmosphere, having any assigned height of the mercury, may 

 most easily be found ; for the line C B in the hyperbola, whose areas denote the 



