Theory of an tniproved Line of Logarithms. 373 



whole fliall interfcft it at divifions which denote numbers in geometrical progrefTion ; then, 

 from the condition of the arrangement, and the property of this logarithmic line, it follows, 

 firft, that every right line fo drawn will, by its interfeftions, indicate a geometrical feries 

 cf numbers * ; fecondly, that fuch feries as are fo indicated by parallel right lines, will 

 have the fame common ratio fj and thirdly, that the feries thus indicated by two parallel 

 right lines, fuppofed to move laterally without changing either their mutual diitance, or 

 parallehfm to themfelves, will have each the fame common ratio ; and, in all pairs of 

 feries indicated by fuch two lines, the ratio between an antecedent on one parallel and the 

 oppoGte term on the other, taken as a confequent, will be conftant J. 



5. In the foregoing paragraphs the logarithmic line has been confidered as unlimrted. On 

 fuch a line, therefore, any antecedent and confequent being given, it would be poflible to 

 find both on the arrangement, and to draw two parallel lines, one over each number : and 

 if the lines be then fuppofed to move without changing either their diftance or aibfolute 

 diredion, fo that the line, which before marked an antecedent, may in tjie fecond ftation 



mark 



* Let AE, CD, EF (Plate XVI. Fig. i.) be portions of the logarithmic line arranged according to the 

 condition ; let GH be a right line drawn acrofs, lb as to pafs through points of divifion e, c, a, denoting num- 

 bers in geometrical progreffion ; then will any other line IK, drawn acrofi the arrangement, alfo pafs through 

 points/, d, b, denoting numbers in geometrical progrellion. 



DEMONSTRATioN. From one of the extreme points of interfeiftion/, in the laft named line IK, draw the 

 light line/^ parallel to GH, and interfetiing the arrangement in the points /, h; and the ratios of the num- 

 bers e ■.f,c -.i, and a : b, will be equal, becaufc the LnterYals on the logarithmic line, or dif&rences of the loga- 

 rithms of thefe numbers, arc equal : 



Ori=/and i = 4. 

 c I ah 



But - = — by the condition. 

 c a ' 



Therefore — = -7 ; or the nnmbers /, «', A, are in the fame continued ratio> as the numbers e, c, a, 

 t h 



Again, the foxMf, the line id, and the line hb, are in arithmetical progreffion, and denote the dilFerences of 



the logarithms of the numbers/ and/, / and d, h and b. 



The quotients of the numbers thcnnfelves are therefore in geometrical progreflion, that ia, 



L L L ' _d± 

 J '' i ' h' °'' 'd~ bi' 



Or T = 77> hy fubftttuting - for its equal r. 



WheiKe-:^ =-7- or /:</:*. Q^E.D, 



i In the fame manner, as it was proved that the line fg parallel to GH pafTes through points of divifon 

 deaoting numbers in the fame continued ratio as thofe indicated by the line GH, it may alfo be (hewn, that 

 the line LM, parallel to any other line IK, will pafs through a feries of numeral points having the fame con- 

 tinued ratio as the feries indicaud by that line IK to which it is paralleL 



* Becaufe the lines prcferve their parallelifm to their former fituation, they will indicate geometrical feries 

 hsving the faroe common ratio as before ; and, becaufe their diftance mcafured on the logarithmic tine remains 

 unchanged, the differences of the logarithms of oppofitc numbers, and confequcnily their ratio, will be conflant. 



