248 Mr. Nicholson’s Method of graduating 
that fuch feries, as are fo indicated by parallel right lines, will 
have the fame common ratio * ; and, thirdly, that the feries 
thus indicated by two parallel right lines, fuppofed to move 
laterally without changing either their mutual diftance, or 
parallel ifm to themfelves, will have each the fame com- 
mon ratio; and in all pairs of feries indicated by fuch two 
lines, the ratio between an antecedent on one parallel and the 
oppofite term on the other, taken as a coniequent,. will be 
conftant -j-. 
5. Thus far the logarithmic line has been confidered as un- 
limited. If, therefore, an antecedent and consequent be given, 
it will be poffible 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 abfolute direction, fo that the line, which before marked an 
antecedent, may now mark a new antecedent ; the other (by 
2. and 3.) will mark a number, at the fame relative pofition 
and diftance, which (hall be the confequent to this laft ante- 
cedent after the fame ratio. 
6. Suppofe a logarithmic line to contain no more than a 
Angle range of numbers from 1 to 10, it will not be nec.effary, 
for the purpofes of computation, to repeat it; for if a Aider 
* In the fame manner as it was proved that the line fg t parallel to GH, palfes 
through points of divifion, denoting numbers in the fame continued ratio as thofe 
indicated by the line GH, it may alfo be fhewn, that the line LM parallel to any 
other line IK, will pafs through a feries of numeral points, having the fame 
continued ratio as the feries indicated by that line IK to which it is parallel. 
f Becaufe the lines preferve their parallelifm to their former fituation, they 
will indicate geometrical feries having the fame common ratio as before; and, 
becaufe their diftance meafured on the logarithmic line remains unchanged,, the 
differenGes of the logarithms of oppoftte numbers, and confequently their ratio, 
will be oonftant, 
3 or ' 
