101 
Note on ‘‘ Nore on SmitH’s DEFINITION OF MULTIPLICATION.” By A. 
L. BAKER. 
The rule should be: To multiply one quantity by another, perform 
upon the multiplicand the series of operations which was performed upon 
unity to produce the multiplier. 
This does not mean, perform upon the multiplicand the series of suc- 
cessive operations which was performed upon unity and upon the suc- 
cessive results. 
Thus, to multiply b by Va: If we attempt to consider Va as derived 
by taking unity a times and then extracting the square root of the result, 
we violate the rule. To get Va by performing operations upon unity, we 
must (e€. g., a=2) take unity 1 time, .4 times, .01 times, .004 times, etc., and 
add the results. Doing this to b, we get the correct result, viz., V2 b= 
1.414...b. 
The rule is thus universal, applying to all multipliers, complex, qua- 
_ternion and irrational. 
THE GEOMETRY OF Simson’'s Line. By C. E. Smirn, [NprIANA UNIVERSITY. 
1. If from any point in the circumference of the circumcircle to a /\ ABC 
1s to the sides of the /\ be drawn, their feet, P,, P,, and P;, lie in a straight 
line. This is known as Simson’s Line. 
(a) First proof that P,, P,, and P, lie in a straight line. 
Since 7s PP, Band PP, B (Fig. 1.) are both right / s, P, P;, P; and B 
are concyclic. 
Likewise P, P,, A, and P, are concyclic. 
INoweA,PP. PB) --*Z PBR. — 80: 
and Z PAC+ Z PBP, = 180°. 
wee ees by PAG: 
But 2 PAC-+ Z PAP, 180°. 
a/b. Pye A 380: 
But Z PAP, = Z PP, P, (measured by same arc of auxiliary circle) 
Ae Poe yj bP 1802 nor arsiraghte7- 
.*. P, P,; and P, lie in a straight line. 
