212 Tktory of an hnfrnvei Line of Logarithms. 



fentiiig both exaggerated and in full contrad, to overwhelm the mind bj furprUe xai 

 alarm ? 



After fuppcr we flroHcd up to the mountain, which now no longer refoundcd with the- 

 confufed noil'c of pickaxes auJ hammers; all was hullied in profound Glencci and th* 

 moon-bearas, which were reflcilcd bright from the fides of the vail excavations, could 

 fcarcely penetrate the deep abyfs bt-low. As we returned we v/ere ftruck with the clear 

 red vivid flames ifluing in a large body from the long range of fmelting-houfes on the coaft, 

 and cafting their rays to a great diftance. 



VIII. 



A .^TitbiJ of dif[^Cj1ng Gunter'i Line of Numbers, by •which the Divift^ns are enlarged, eitid 

 other Advantages obtained. 



G, 



'F the many ingenious inflruments for computation which were in ufe during the laft 

 century, among mathematicians, fcarcely any are to be found at prefcnt except the fedlor 

 and the logarithmic line of Gunter. The eafe and accuracy of computation by thofe ad^ 

 Diirable numbers have rendered the others of little importance ; but the fedor has main- 

 tained its ftation from its utility in graphical operations, and the Guntei's line is not only 

 of great value in nautical and other proportions which do not exceed three places of figures, 

 but alfo as a check to adure the truth of the leading figures in more extended calculations. 

 Aboat ten years ago 1 communicated to the Royal Society a method of extending the range 

 of this laft inftrument, which I (ILll confider as lefs generally known, than its utility may 

 perhaps claim. The principles depend on the following conGderations : 



1. If two geometrical feries of numbers, having the fame common ratio, be placed in or- 

 der with the terms oppofite each other, the ratio between any term in one feries and its 

 oppofite in the other will be conftant *. 



2. And the ratio of a term in one feries to any term in the other, will be the fame as ob- 

 tains between any other two terms having the fame relative pofition and diftance f. 



3. In all fuch pairs of geometrical feries as have the fame common ratio, the laft-men- 

 tioned property obtains, though tlie iiril antecedent and confequent be taken in one pair, 

 and the fecond in any other pair :f. 



4. If the dilTerences of the logarithms of ntunbers be laid in order upon an arrangement 

 fif eq;tl-di(lant parallel right lines, in fuch a manner as that a right line draw" an-ofs tbj: 



* G*"""- '"'" \b bn bn- bn' b„* 



Then a : 6 :: an : ba :: an' : bn*, &c. 

 f In the foregoing feries a : in' : : an* : ba* : : an : bn*, &c. 

 . /^ _ r • fa an an'' an' an* 



fd dn dn' dn* dn* 

 GeoiDi feries < bd bdn bdn* bdn^ idn* 

 *-a a a a a 



Then a t iii' a dn : , Sec. 



