226 Proceedings of the Royal Society 
The rate of diminution of SP is evidently 
to COS a 
PT. COS a = 
SP = 5 SP ; 
sm a 2 ’ 
that is, SP diminishes in geometrical progression as time increases, 
h 
the rate being ^ per unit of time per unit of length. By an 
ordinary result of arithmetic (compound interest payable every 
7c 
instant) the diminution of log. SP in unit of time is ~ * 
A 
This process of solution is only applicable to resisted harmonic 
h 
vibrations when n is greater than When n is not greater than 
Jc 
2 the auxiliary curve can no longer be a logarithmic spiral, for the 
moving particle never describes more than a finite angle about the 
pole. A curve, derived from an equilateral hyperbola, by a process 
somewhat resembling that by which the logarithmic spiral is 
deduced from a circle, must be introduced ; and then the geo- 
metrical method ceases to be simpler than the analytical one, so 
that it is useless to pursue the investigation farther, at least from 
this point of view. 
3. On the Antiquity of Intellectual Man : from a Practical 
and Astronomical point of view. By Professor C. Piazzi 
Smyth. 
The author of this paper confined himself entirely to the use of 
contemporary data ; and amongst these, the only kinds found avail- 
able for the very earliest ages, were buildings of one kind or 
another. 
Examining these again, after their first approximate arrange- 
ment in chronological relative order by well-known archaeologists, 
it was discovered that there resided a peculiar astronomical opposi- 
tion between the chief structures in the valley of the Euphrates 
and those in the valley of the Nile. 
The final contest for antiquity amongst all the archaeological 
known remains of the whole earth seemed to lie entirely with the 
