415 
years and experience in this errantry, they purchase their free- 
dom by some tryall of skill in y* faculty w™ they perform in 
publick before y° Majistrates of y° place, w™ is testifyed by 
an instrument under y° seale of y* magistracy. I believe if 
we should deny freedom to all such as leave y* own country 
and come to plant among us, we should doe y™ noe injury, 
for none of y™ having undergone this tryall, they would be 
noe better y" journeymen at home, but by our naturall civility 
for strangers has our law run more in y" fayor.’” 
Sir William Rowan Hamilton read a Paper on a new 
System of Roots of Unity, and of operations therewith con- 
nected: to which system of symbols and operations, in conse- 
quence of the geometrical character of some of their leading 
interpretations, he is disposed to give the name of the “ Ico- 
sian CaLcuULus.” 
This Calculus agrees with that of the Quaternions, in 
three important respects: namely, Ist, that its three chief 
symbols, «, x, A, are (as above suggested) roots of unity, as 
2, j, k are certain fourth roots thereof: 2nd, that these new 
roots obey the associative law of multiplication; and 3rd, that 
they are not subject to the commutative law, or that their 
places as factors must not in general be altered in a product. 
And it differs from the Quaternion Calculus, Ist, by involv- 
ing roots with different exponents; and 2ndly, by not re- 
quiring (so far as yet appears) the distributive property of 
multiplication. In fact, + and -, in these new calculations, 
enter only as connecting exponents, and not as connecting terms : 
indeed, no terms, or in other words, no polynomes, nor even 
binomes, have hitherto presented themselves, in these late 
researches of the author. As regards the exponents of the 
new roots, it may be mentioned that in the principal system,— 
for the new Calculus involves a family of systems,—there are 
adopted the equations, — 
bea@= =X X= «5 (A) 
VOL. VI. 2a 
