Notation

The table of the algorithm organises each command in order. Each command denotes one operation being made on two terms. The second column states only the operator being used. Variables are notated as "${\displaystyle V}$",^{[lower-alpha 3]} where the superscript before it represents the amount of different values the variable has been assigned to, and the subscript after it represents the ordinal assignment of the variable, that is which variable it is. (For example, ${\displaystyle ^{2}V_{4}}$ refers to the second assignment of variable number 4. Any variables hitherto undefined have a superscript of 0.) The variables are numbered starting from ${\displaystyle V_{0}}$. The third column tells the computer exactly what command is taking place, (For example, on line 1, the command performed is "${\displaystyle ^{1}V_{2}\times ^{1}V_{3}}$" - the first iteration of variable 2 is multiplied by the first iteration of variable 3.) and only incorporates one operation between two terms per line. Column 4 - "Variables receiving results" takes note of where the result of the operation in column 3 should be stored. In this way, any variables in this column have their superscript number incremented by one each time. (e.g. on line 1, the result of ${\displaystyle ^{1}V_{2}\times ^{1}V_{3}}$ is assigned to variables ${\displaystyle V_{4}}$, ${\displaystyle V_{5}}$, and ${\displaystyle V_{6}}$.)

Column 5 states whether either of the variables used in the operation of the command has been changed. Enclosed in curly braces, two rows per command put the original variable on the left side of an equals sign, and the new variable on the other side - that is, if the variable has been changed, its superscript is incremented by one, and if not, it remains the same. (e.g. line three assigns the result of ${\displaystyle ^{1}V_{5}+^{1}V_{1}}$ to the second iteration of the variable ${\displaystyle V_{5}}$, and the fifth column reflects this by noting;

${\displaystyle {\begin{Bmatrix}^{1}V_{5}=^{2}V_{5}\\^{1}V_{1}=^{1}V_{1}\end{Bmatrix}}}$

${\displaystyle V_{5}}$ has changed, but ${\displaystyle V_{1}}$ hasn't.

In column 6, "Statement of Results", the result assigned to the variable in column 4 is shown in its exact value based on the values of the two terms previously assigned. (e.g. on line 1 - ${\displaystyle ^{1}V_{2}\times ^{1}V_{3}}$ - ${\displaystyle V_{2}}$ was set at the beginning to be ${\displaystyle 2}$, and ${\displaystyle V_{3}}$ was set to be the variable ${\displaystyle n}$. Therefore, ${\displaystyle ^{1}V_{2}\times ^{1}V_{3}=2n}$, in mathematical notation.) This column is ostensibly not computed by the engine, and appears to be more to aid clarity and the reader's ability to follow the steps of the program. (For example, line 5 has a fraction being divided by two, which is notated as it being multiplied by a half, probably for coherence and the typographical complexity of a nested fraction.) It also makes use of separate variable notation outside of the program, the ${\displaystyle A}$ and ${\displaystyle B}$ variables, which are multiplied successively to find the final value, ${\displaystyle B_{7}}$, thus:^[10]

${\displaystyle B_{7}=-1(A_{0}+B_{1}A_{1}+B_{3}A_{3}+B_{5}A_{5})}$

Beyond this, each successive column shows the values of a given variable over time. Each time a variable either changes, or has its value become relevant by token of its presence as one of the terms in the current command, its value is stated or restated in its respective column. Otherwise, it is marked with an ellipsis to denote its irrelevancy. This presumably mimics the computer's need for only relevant information, thereby tracking the value of a variable as the program parses.^[10]

Method

The program sought to calculate what is known by modern convention as the eighth Bernoulli number, listed as ${\displaystyle B_{7}}$, as Lovelace begins counting from ${\displaystyle B_{0}}$.^[24]

Error

In operation 4, the division supposedly taking place is "${\displaystyle ^{2}V_{5}\div ^{2}V_{4}}$", to be stored in variable ${\displaystyle ^{1}V_{11}}$. However, the "Statement of results" says that the division should be:

${\displaystyle {\frac {2n-1}{2n+1}}}$

As a matter of fact, the division is the wrong way round; ${\displaystyle 2n-1}$ is the second iteration of ${\displaystyle V_{4}}$, as can be seen in operation 2. Likewise, ${\displaystyle 2n+1}$ is the second iteration of ${\displaystyle V_{5}}$, as can be seen in operation 3. Thus, operation 4 should not be ${\displaystyle ^{2}V_{5}\div ^{2}V_{4}}$, but rather ${\displaystyle ^{2}V_{4}\div ^{2}V_{5}}$. This bug means that if the engine were ever to run this algorithm in this state, it would fail to generate Bernoulli numbers correctly, and would find its final goal value (the eighth Bernoulli number, ${\displaystyle -{\tfrac {1}{30}}}$) to be ${\displaystyle -{\tfrac {25621}{630}}}$.

Modern implementations

Lovelace's program can be implemented in a modern programming language, though due to the above stated error, if transcribed exactly it would return an incorrect final value for ${\displaystyle B_{7}}$. The original program generalised in pseudocode follows as thus:

V[1] = 1
V[2] = 2
V[3] = n (n = 4 in Lovelace's program.)

V[4] = V[4] - V[1]
V[5] = V[5] + V[1]
V[11] = V[5] / V[4]
V[11] = V[11] / V[2]
V[13] = V[13] - V[11]
V[10] = V[3] - V[1]
V[7] = V[2] + V[7]
V[11] = V[6] / V[7]
V[12] = V[21] * V[11]
V[13] = V[12] + V[13]
V[10] = V[10] - V[1]
V[6] = V[6] - V[1]
V[7]= V[1] + V[7]
//Finish Later

The implementation in pseudocode highlights the fact that computer languages define variables on a stack, which obviates the need for tracking and specifying the current iteration of a variable. In addition, Lovelace's program only allowed for variables to be defined by performing addition, subtraction, multiplication or division on two terms that were previously defined variables. Modern syntax would be capable of performing each calculation more concisely. This restriction becomes apparent in a few places, for example on command 6 (${\displaystyle V_{13}=V_{13}-V_{11}}$). Here Lovelace defines a hitherto undefined variable (${\displaystyle V_{13}}$) by itself, thereby assuming that all undefined variables are automatically equal to 0, where most modern programming languages would return an error or list the variable as null. What she intended was "${\displaystyle 0-V_{11}}$", but had constrained herself to only using variables as terms. Likewise, in command 8 (${\displaystyle V_{7}=V_{2}+V_{7}}$), the strict notation of two-term arithmetic becomes cumbersome, as in order to define ${\displaystyle V_{7}}$ as 2, Lovelace assigns its value (0) to itself plus ${\displaystyle V_{2}}$ (2). It is due to this restrictive notation that ${\displaystyle V_{7}}$ is defined thus.