equations of all orders, by continuous approximation. 317 
at their first occurrence in the table, and their ultimate appli- 
cation at the second. The operations included in the paren- 
theses may be mentally effected, whenever r is a simple digit. 
And lastly, the vertical arrangement of the addends adapts 
them at once to the purposes of arithmetic, on every scale of 
notation. 
General Synopsis. 
n— 2 ,4 Derivee. 
n 3'^Derivee. 
3rd. Derivee. 
Dr—) D 
0 
2nd. Derivee. 
V 
(Ur=) U 
O 
t 
(rV — ) V 
O 
&c. 
b 
(Cr = ) C 
O 
V 
T 
&c. 
C 
B 
(U + r 2 =) U 
0 X 
(V + U r~) V 
0 I I 
(D + Er — ) D 
O I I 
(C + Dr-) C 
0 I 1 
(U+r 1 -) U 
I 2 
(V + Ur = ) V 
12 2 
(D+Er— ) D 
12 2 
(C + Dr=) C 
I 2 2 
&c. to 
U 
See. to 
V 
Sec. 
(D+Er=) D 
2 3 3 
(DV=) D' 
0 
(C’r'zz) C' 
O 
(U' /=) V 
O 
(V 1 r'-) V 
0 
B' 
(C'+DV=)C' 
0 1 ,i 
&c. 
C' 
V' 
(U'-j-r i2 — ) U 
0 I 
& C. 
T' 
(V+UV=)V' 
OI X 
&c. 
&C. 
(D'4-E'r'— )D' 
0 1 1 
&e. 
& c. 
1 st. Derivee. 
(B/~) B 
O 
Synthesis. 
<pR 
(A r=) A 
O 
Anal. 
A 
-A 
O 
A 
tp R 
A' 
(B-fCrr) B' 
< 
Jl 
< 
—A' 
O I I 
0 
O 
(B V=) B' 
O 
— 
<pR" 
A' 
(A";*— )A" 
—A" 
(B' + C / r / =:)B' 
O 
O 
0 1 I 
&c. 
&c. 
Root. 
R •+■ r + r' + . 
Illustrations. 
19. The remarks which are yet to be adduced will bear 
almost exclusively on the Analytic portion of the Theorem, 
from which the Synthetic differs only in the less intricate 
management of the first derivee ; this function having no 
concern with the discovery of the root, and its multiple being 
additive like all the rest, instead of subtractive. 
From the unrestricted nature of the notation employed, it 
is evident that no class of equations, whether finite, irrational 
or transcendental, is excluded from our design. In this re- 
spect indeed, the new method agrees with the established 
