(Beginning of) The First Rule of Logic, by C. S. Peirce

[Matt Faunce]

Here are the first two pages of The First Rule of Logic, by C. S. Peirce

“Certain methods of mathematical computation correct themselves; so that if
an error be committed, it is only necessary to keep right on, and it will
be corrected in the end. For instance, I want to extract the cube root of
2. The true answer is 1.25992105… . The rule is as follows:

“Form a column of numbers, which for the sake of brevity we may call the
A’s. The first three A’s are any three numbers taken at will. To form a new
A, add the last two A’s, triple the sum, add to this sum the last A but
two, and set down the result as the next A. Now any A, the lower in the
column the better, divided by the following A gives a fraction which
increased by 1 is approximately [the cube root of two.]

“You see the error committed in the second computation, though it seemed to
multiply itself greatly, became substantially corrected in the end.

“If you sit down to solve ten ordinary linear equations between ten unknown
quantities, you will receive materials for a commentary upon the
infallibility of mathematical processes. For you will almost infallibly get
a wrong solution. I take it as a matter of course that you are not an
expert professional computer. He will proceed according to a method which
will correct his errors if he makes any.”

[Because of the limitations of Usenet, Peirce’s table is re-represented by
me, as follows:]

Table 1

Correct Computation

{A + previous A} {tripled} {+2nd from last A}

{1}
{0}
{1 + 0 = 1} {1 × 3 = 3} {3 + 1 =}
{4 + 1 = 5} {5 × 3 = 15} {15 + 0 =}
{15 + 4 = 19} {19 × 3 = 57} {57 + 1 =}
{58 + 15 = 73} {73 × 3 = 219} {219 + 4 =}
{223 + 58 = 281} {281 × 3 = 843} {843 + 15 =}
{858} (and so on)
{3301}
{12700}
3301 ÷ 12700 + 1 = 1.2599213
[More accurate answer = 1.25992104989]
Error = +.0000002+

Same computation but with the fifth A in error, marked by ! :

{1}
{0}
{1 + 0 = 1} {1 × 3 = 3} {3 + 1 =}
{4 + 1 = 5} {5 × 3 = 15} {15 + 0 =}
{16! + 4 = 20} {20 × 3 = 60} {60 + 1 =}
{61} (and so on)
{235}
{904}
{3478}
{13381}
3478 ÷ 13381 + 1 = 1.25992078
[More accurate answer = 1.25992104989]
Error = -.0000002+

“This calls to mind one of the most wonderful features of reasoning, and
one of the most important philosophemes in the doctrine of science, of
which however you will search in vain for any mention in any book I can
think of, namely, that reasoning tends to correct itself, and the more so
the more wisely its plan is laid. Nay, it not only corrects its
conclusions, it even corrects its premises. The theory of Aristotle is that
a necessary conclusion is just equally as certain as its premises, while a
probable conclusion somewhat less so. Hence, he was driven to his strange
distinction between what is better known to Nature and what is better know
to us. But were every probable inference less certain than its premises,
science, which piles inference upon inference, often quite deeply, would
soon be in a bad way. Every astronomer, however, is familiar with the fact
that the catalogue place of a fundamental star, which is the result of
elaborate reasoning, is far more accurate than any of the observations from
which it was deduced.”

[Matt Faunce]

Matt Faunce <mattf...@gmail.com> wrote:
> Here are the first two pages of The First Rule of Logic, by C. S. Peirce
>
> “Certain methods of mathematical computation correct themselves; so that if
> an error be committed, it is only necessary to keep right on, and it will
> be corrected in the end. For instance, I want to extract the cube root of
> 2. The true answer is 1.25992105… . The rule is as follows:
>
> “Form a column of numbers, which for the sake of brevity we may call the
> A’s. The first three A’s are any three numbers taken at will. To form a new
> A, add the last two A’s, triple the sum, add to this sum the last A but
> two, and set down the result as the next A. Now any A, the lower in the
> column the better, divided by the following A gives a fraction which
> increased by 1 is approximately [the cube root of two.]
>
> “You see the error committed in the second computation, though it seemed to
> multiply itself greatly, became substantially corrected in the end.
>
> “If you sit down to solve ten ordinary linear equations between ten unknown
> quantities, you will receive materials for a commentary upon the
> infallibility of mathematical processes. For you will almost infallibly get
> a wrong solution. I take it as a matter of course that you are not an
> expert professional computer. He will proceed according to a method which
> will correct his errors if he makes any.”
>
> [Because of the limitations of Usenet, Peirce’s table is re-represented by
> me, as follows:]
>

[Here’s the table again, but double spaced this time. (My crappy
newsreader, NewsTap, doesn’t recognize single-spaced paragraph returns.)]

--
Matt