The New Method of Evaluation as Applied to Pi
The New Method of Evaluation as Applied to Π
by Lewis Carroll
- "Little Jack Horner
- sat in a corner
- eating his Christmas pie."
The problem of evaluating π which has engaged the attention of mathematicians from the earliest ages, had, down to our own time, been considered as purely arithmetical. It was reserved for this generation to make the discovery that it is in reality a dynamical problem: and the true value of π which appeared an ignis fatuus to our forefathers, has at last been obtained under pressure.
The following are the main data of the problem:
Let U = the University, G = Greek, and P = Professor. Then GP = Greek Professor; let this be reduced to its lowest terms, and call the result J.
Also let W = the work done, T = the Time, p = the given payment, π = the payment according to T, and S = the sum required; so that π = S.
The problem is, to obtain a value for π which shall be commensurable with W.
In early treatises on this subject, the mean value assigned to π will be found to be 40.000000. Later writers suspected that the decimal point had been accidentally shifted, and that the proper value was 400.00000: but, as the details of the process for obtaining it had been lost, no further progress was made in the subject until our own time, though several most ingenious methods were tried for solving the problem.
Of these methods we proceed to give some brief account. Those chiefly worthy of note appear to be Rationalization, the Method of Indifferences, Penrhyn's Method, and the Method of Elimination.
We shall conclude with an account of the great discovery of our own day, the Method of Evaluation under Pressure.
The peculiarity of this process consists in its affecting all quantities alike with a negative sign.
To apply it, let H = High Church, and L = Low Church, then the geometric mean = √HL: call this "B" (Broad Church).
- ∴ HL = B².
Also let x and y represent unknown quantities.
The process now requires the breaking up of U into its partial fractions, and the introduction of certain combinations. Of the two principal factions thus formed, that corresponding with P presented no further difficulty, but it appeared hopeless to rationalize the other.
A reductio ad absurdum was therefore attempted, and it was asked "why should π not be evaluated?" The great difficulty now was, to discover y.
Several ingenious substitutions and transformations were then resorted to, with a view to simplyfying the equation, and it was at one time asserted, though never actually proved, that all the y's were on one side. However, as repeated trials produced the same irrational result, the process was finally abandoned.
II. The Method of Indifferences
This was a modification of "the method of finite Differences," and may be thus briefly described--
Let E = Essays, and R = Reviews: then the locus of (E + R), referred to multilinear coördinates, will be found to be a superficies (i.e., a locus possessing length and breadth, but no depth). Let v = novelty, and assume (E + R) as a function of v.
Taking this superficies as the plane of reference, we get--
- E = R = B
- ∴ EB = B² = HL (By the last article)
Multiplying by P, EBP = HPL.
It was now necessary to investigate the locus of EBP: this was found to be a species of Catenary, called the Patristic Catenary, which is usually defined as "passing through origen, and containing many multiple points." The locus of HPL will be found almost entirely to coincide with this.
Great results were expected from the assumption of (E + R) as a function of v: but the opponents of this theorem having actually succeeded in demonstrating that the v-element did not even enter into the function, it appeared hopeless to obtain any real value of π by this method.
III. Penrhyn's Method
This was an exhaustive process for extracting the value of π in a series of terms, by repeated divisions. The series so obtained appeared to be convergent, but the residual quantity was always negative, which of course made the process of extraction impossible.
This theorem was originally derived from a radical series in Arithmetical Progression: let us denote the series itself by A.P., and its sum by (A.P.)S. It was found that the function (A.P.)S. entered into the above process, in various forms.
The experiment was therefore tried of transforming (A.P.)S. into a new scale of notation: it had hitherto been, through a long series of terms, entirely in the senary, in which scale it had furnished many beautiful expressions: it was now transferred into the denary.
Under this modification, the process of division was repeated, but with the old negative result: the attempt was therefore abandoned, though not without a hope that future mathematicians, by introducing a number of hitherto undetermined constants, raised to the second degree, might succeed in obtaining a positive result.
IV. Elimination of J
It had long been perceived that the chief obstacle to the evaluation of π was the presence of J, and in an earlier age of mathematics J would probably have been referred to rectangular axes, and divided into two unequal parts-- a process of arbitrary elimination which is now considered not strictly legitimate.
It was proposed, therefore, to eliminate J by an appeal to the principle known as "the permanence of equivalent formularies": this, however, failed on application, as J became indeterminate. Some advocates of the process would have preferred that J be eliminated in toto. The classical scholar need hardly be reminded that toto is the ablative of tumtum, and that this beautiful and expressive phrase embodied the wish that J should be eliminated by the compulsory religious examination.
It was next proposed to eliminate J by means of a canonisant. The chief objection to this process was, that it would raise J to an inconvieniently high power, and would after all only give an errational value for π.
Other processes, which we need not here describe, have been suggested for the evalution of π. One was, that it should be treated as a given quantity: this theory was supported by many eminent men, at Cambridge and elsewhere; but on application, J was found to exhibit a negative sign, which of course made the evaluation impossible.
We now proceed to describe the modern method, which has been crowned with brilliant and unexpected success, and may be defined as
V. Evaluation Under Pressure
Mathematicians had already investigated the locus of HPL, and had introduced this function into the calculation, but without effecting the desired evaluation, even when HPL was transferred to the opposite side of the equation, with a change of sign. The process we are about to describe consists chiefly in the substitution of G for P, and the application of pressure.
Let the function φ (HGL) be developed into a series, and let the sum of this be assumed as a perfectly rigid mody, moving in a fixed line; let "μ" be the coefficient of moral obligation, and "e" the expediency. Also let "F" be a Force acting equally in all directions, and varying inversely as T: let A = Able, and E = Enlightened.
We have now to develop φ (HGL) by Maclaurin's Theorem.
The function itself vanishes when the variable vanishes:
φ (o) = O φ' (o) = C (a prime constant) φ'' (o) = 2.J. φ'' ' (o) = 2.3.H φ'' '' (o) = 2.3.4.S φ' '' ' (o) = 18.104.22.168.P φ'' '' '' (o) = 22.214.171.124.6.J
after which the quantities recur in the same order.
The above proof is taken from the learned treatise "Augusti de fallibilitate historicorum," and occupies an entire Chapter: the evaluation of π is given in the next Chapter. The author takes occasion to point out several remarkable properties, possessed by the above series, the existence of which had hardly been suspected before.
This series is a function both of μ and of e: but, when it is considered as a body, it will be found that μ = 0 and that e only remains.
We now have the equation
- φ (HGL) = O + C + J + H + S + P + J.
The summation of this gave a minimum value for π: this, however, was considered only as a first approximation, and the process was repeated under pressure EAF, which gave to π a partial maximum value: by continually increasing EAF, the result was at last obtained.
- π = S = 500.00000.
The result differs considerably from the anticipated value, namely 400.00000: still there can be no doubt that the process has been correctly performed, and that the learned world may be congratulated on the final settlement of this most difficult problem.