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v 
the Surface generated by the Perimeter of the Curve, 
the fpirai Area generated by the revolving Ray, the 
fpiral Line itfelf, or the Sum of the Terms of a Pro- 
grefiion, have fuch Limits or not 5 and for mca- 
liiring thofe Limits. The Author infills on thefe 
Subjeds, the rather that they are commonly deferibed 
in very myfterious Terms, and have been the moil 
fertile of Paradoxes of any parts of the higher Geo- 
metry. Thefe Paradoxes, however, amount to no 
more than this : That a Line or Number may be con- 
tinually acquiring Increments, and thofe Increments 
may decrcale in fuch a manner, that the whole Line 
or Number fhall never amount to a given Line or 
Number. The Neceflity of admitting this is obvious 
enough, and is here fhewn from the Nature of the 
moft common geometrical Figures in Art. 292, 293, 
&c. and from any Series of Fractions that decreafe 
continually, in Art. 3 54 > 3 5 5 > &c . 
The Xlth Chapter treats of the Curvature of Lines, 
its Variation, the Degrees of Contad of the Curve 
and Circle of Curvature, and of various Problems 
that depend on the Curvature of Lines. This Subject 
is treated fully, becaufe of its extenfive Ufefulnefs, and 
becaufe in this confifts one of the greateft Advantages 
of the modern Geometry above that of the Antients. 
The Author on this, as former Occafions, begins by 
premifing the neceflary Definitions. Curve Lines 
touch each other in a Point, when the fame right 
Line is their common Tangent at that Points and 
that which has the elofeft Contad with the Tangent, 
or paffes betwixt it and the other Curve through the 
Angle of Contad formed by them, being lefs infleded 
from the Tangent, is therefore lefs curve. Thus a 
greater 
