[ f 61 ] 
This will be evident, by fuppofing the Radii fo 
near to one another, that the intercepted Parts of 
the Spiral may be taken as right Lines : For then 
there will be a Scries of fimilar Triangles, each hav- 
ing an equal Angle at the Centre, and the Sides about 
thofc Angles proportional. 
Art. II. The fame things fill fuppo fed, the ‘Tarts 
of the Circumference of the Circle , reckoned from 
any one Toint , may be taken as the Logarithms 
of the Ratio’s between the correfponding Rays of 
the S r i al. 
For thofc Rays are a Scries of Terms in a conti- 
nued geometric Progrellion ; and the Parts of the 
Circumference form a Series of Terms in arithme- 
tic Progrellion. Now the Terms of the arithmetic 
Series being taken as the Exponents of the corre- 
fponding Terms in the geometric Series, there will 
be the fame Relation between each geometric Term 
and its Correlative, as between Numbers and their 
Logarithms. And hence the proportional Spiral is 
alfo called the logarithmic Spiral. 
A r 1. 1 1 1. That proportional Spiral , which interfeffs 
its Radii at Angles of Degrees , produces Lo- 
garithms that are 0 / NapierT Kind. 
For, it the Difference between the frit and fecond 
Terms in the geometric Series was indefinitely fmall, 
and the fir If Divifion of the Circumference was of 
the fame Magnitude, then may that Part of the Spi- 
ral, intercepted between the firft and fecond Radii, 
be taken as the Diagonal of a Square, twoofwhofe 
Sides arc Parts of thofe Radii : Therefore the Spiral 
which 
