Page:Scientific Memoirs, Vol. 3 (1843).djvu/713

Each of the squares below the zeros is intended for the inscription of any general symbol or combination of symbols we please; it being understood that the number represented on the column immediately above, is the numerical value of that symbol, or combination of symbols. Let us, for instance, represent the three quantities $\scriptstyle {a}$, $\scriptstyle {n}$, $\scriptstyle {x}$, and let us further suppose that $\scriptstyle {a=5}$, $\scriptstyle {n=7}$, $\scriptstyle {x=98}$. We should have—

$\scriptstyle {\mathbf {V} _{1}}$	$\scriptstyle {\mathbf {V} _{2}}$	$\scriptstyle {\mathbf {V} _{3}}$	$\scriptstyle {\mathbf {V} _{4}}$	&c.
+[1]	+	+	+
0	0	0	0
0	0	0	0	&c.
0	0	9	0
5	7	8	0	&c.
$\scriptstyle {a}$	$\scriptstyle {n}$	$\scriptstyle {x}$

We may now combine these symbols in a variety of ways, so as to form any required function or functions of them, and we may then inscribe each such function below brackets, every bracket uniting together those quantities (and those only) which enter into the function inscribed below it. We must also, when we have decided on the particular function whose numerical value we desire to calculate, assign another column to the right-hand for receiving the results, and must inscribe the function in the square below this column. In the above instance we might have any one of the following functions:—

$\scriptstyle {ax^{n},\qquad x^{an},\qquad a\cdot n\cdot x,\qquad {\frac {a}{n}}x,\qquad a+n+x,}$ &c. &c.

Let us select the first. It would stand as follows, previous to calculation:—

$\scriptstyle {\mathbf {V} _{1}}$	$\scriptstyle {\mathbf {V} _{2}}$	$\scriptstyle {\mathbf {V} _{3}}$	$\scriptstyle {\mathbf {V} _{4}}$	&c.
+	+	+	+
0	0	0	0	&c.
0	0	9	0
5	7	8	0	&c.
$\scriptstyle {a}$	$\scriptstyle {n}$	$\scriptstyle {x}$	$\scriptstyle {ax^{n}}$	&c.
$\scriptstyle {\underbrace {\quad \quad \quad \quad \quad } }$
$\scriptstyle {ax_{n}}$

The data being given, we must now put into the engine the cards proper for directing the operations in the case of the particular function chosen. These operations would in this instance be,—

Firstly, six multiplications in order to get $\scriptstyle {x^{n}}$ ($\scriptstyle {=98^{7}}$ for the above particular data).

Secondly, one multiplication in order then to get $\scriptstyle {a\cdot x^{n}}$ ($\scriptstyle {=5.98^{7}}$).

In all, seven multiplications to complete the whole process. We may thus represent them:—

(×, ×, ×, ×, ×, ×, ×), or 7(×).

The multiplications would, however, at successive stages in the solution of the problem, operate on pairs of numbers, derived from different columns. In other words, the same operation would be performed

1. It is convenient to omit the circles whenever the signs + or − can be actually represented.

vol. iii. part xii.

3a