P. HISTORY OF ' AATHEMATICAL - School of Mathematics

INDIVIDUAL WRITERS 225
lines, but because lines drawn one above others often occupy too
much space, and for other reasons, it is often more convenient to
introduce commas and parentheses. Thus a, b+c is the same as
a, b+c or a(b+c); and Gb, c+d is the same as a+b, c+d, or (a+b)
(c+d), i.e., +a+b multiplied by c+d. And, similarly, vinculums are
placed under vinculums. For example, a, bc+ef- is expressed also
thus, a(bc+e(f+g)), and a, bc+ef+9+hlm,n may be written also
+ (a(bc+e(f+g))+hlm)n. What relates to vinculums in multiplication
applies to vinculums in division. For example,
a h
b -+- e +I -
G+3!--?
n
may be written in one line thus:
(a: ((b:c)+(e:, f+g))+h:(l:m)):n ,
and there is no difficulty in this, as long as we observe that whatever
-
fills up a given parenthesis be taken as one quantity. . . . . The same
is true of vinculums in the extraction of roots. Thus da4+de, fq
is the same as d(a4+d(e(f+g))) or d(a4+d(e, f+g)). And
for d'a+bdcc+dd
one may write I/(aa+bd(cc+dd)) :, e+
~+\~j/gg+hh+kk
/(j/(gg+hh)+kk). Again a=b signifies that a is equal to b, and
a-b signifies that a is greater than b, and a=-b that a is less than b.
Also proportionality or analogia of quantities, i.e., the identity of ratio,
may be represented; we may express it in the calculus by the sign of
equality, for there is no need of a special sign. Thus, we may indi-
a
cate that a is to b as 1 is to m by a:b=l:m, i.e., -91 The sign for
b m'
continued proportion is +, so that % a, b, c, and d are continued proportional~.
"There is adopted a sign for similitude; it is v, ; also a sign for
both similitude and equality, or a sign of congruence, E accordingly,
DEF rn PQR signifies that the two triangles are similar; but DEF
PRQ marks their congruence. Hence, if three quantities have to one
another the same ratio that three others have to one another, we may
mark this by a sign of similitude, as a; b; c m 1; m; n means that a is to
b as 1 is to m, and a is to c as 1 is to n, and b is to c as m is to n. . . . . 1t
In the second edition of the Miscellanea Berolznensia, of the year