a es 
, 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, wis 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’ favor.’ ” 
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 CALcULus.” 
This Calculus agrees with that of the Quaternions, in 
ee tC err 
three important respects: namely, Ist, that its three chief 
symbols, 1, x, A, are (as above suggested) roots of unity, as 
i, 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, 1st, by involy- 
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, — 
l=®=68=,A=ck3 (A) 
VOL. VI. 2a 
