performing the involution and evolution of numbers. 27 
as they recede from the centre, and each turn will carry on 
the powers to an exponent 10 times higher than the preceding: 
and the converse will obtain with regard to the descending 
portion. Thus, immediately in a line with the 10, on the supe- 
rior branch of the spiral, is found the number 10000000000, 
or 10 10 : below it on the inferior branches, we find succes- 
sively 1.258926 = 10 0,1 , 1.023293 = io°' 01 , 1.00230524 = 
io‘ 001 , 1.000230285= io' 0001 , 1.0000230261 = io' 00001 ’ &c. 
of which, agreeably to the remarks that were formerly made, 
the decimal figures approach nearer and nearer to 2.302585093, 
&c. the reciprocal of the modulus of the logarithmic system. 
A much greater extension might be given to the scale, by 
multiplying the number of turns of the spiral corresponding 
to the decuple increase of the exponents : but the superior 
accuracy thus obtained would probably be overbalanced by 
the diminished conveniency of application. 
It is possible to exhibit in one view the whole series of 
roots, powers, and exponents, in all their possible relations, 
by the following disposition of lines. Let the lines AB, AC, 
( PL IV.), which I shall call respectively the line of exponents, 
and the line of roots, be drawn at right angles to each other, 
and a diagonal AD, or line of powers be drawn, bisecting this 
angle. Divide AB logometrically, so that the unit of the 
scale shall be at A : upon the same scale, divide AC into logo- 
metric logarithms, and AD into similar parts, by perpendi- 
culars from the divisions of AC. 
Through all these points of division, let there be drawn 
perpendiculars to the respective lines : and let each of these 
perpendiculars be considered as referring always to the num- 
bers on the lines from which they are drawn. The following 
E 2 
