THE BUILDER JUNE 1929

A Consideration of Some of the Difficulties of Squaring the Circle

By BRO. CHARLES H. MERZ Ohio

THIS paper is not intended as an attempt to set aside the commonly accepted ratio or any other, or to uphold the
same; but it is written to set out, so far as opportunity permits, some of the difficulties attendant upon quadrature
of the circle, and to show the rudiments of the complicated and tedious method commonly adopted of attempting
the approximation of the true ratio between the diameter and the circumference.

The different methods of solving the problem of the quadrature of the circle are more than a hundred in number.
The ratios of the circumference to the diameter are equally numerous. Some differences exceed eight hundredths
of the circumference, while others vary as much as twenty-three hundredths.

In the mechanic arts the ratio of the diameter to the circumference is assumed to be as 7 to 22, which is accurate
enough for many purposes, though it is claimed that the real ratio can never be exactly expressed in numbers. In
ordinary mathematical work it is assumed to be as 1 to 3.14159215. One English mathematician has carried out the
decimal to 607 places.

The ratio between the diameter and circumference is fundamental, and any error made in the beginning is carried
into all the operations which depend upon it, and the same is true of any other possible errors that may occur in
the additional operations to ascertain the relation between the circle and the square.

It is of interest to note that the Masonic apron is actually an ancient Egyptian mathematical problem, based upon
the principles of the Operative Mason's Square. showing a quick and very nearly perfect manner of determining
a squared circle, in which the peripheries of both square and circle are of precisely equal length. Correctly analyzed,
it consists of two oblongs of 3 x 4 (at the top) and two oblongs of 4 x 5 (at the bottom).

These constitute a perfect square. Setting one leg of the compasses upon the intersection of the lines that divide the
square and the free leg on A or B. we have a circle the circumference of which is equal to that of the square. Lines
drawn from A and B to E will be of precisely the same length as the distance from E to F which is the vertical axis
of the triangle E-C-D. The relation of this to the circle squaring problem is that A-E, B-E and F-E are the radii of
a circle of almost equal perimeter to the whole square. In the square taken as the base and the triangle E-C-D as
the vertical section thereof, we have the precise geometrical proportions of the Great Pyramid of Gizeh.


As there are comparatively few Masons who are familiar with that most admirable work written by the late Brother
H. P. H. Bromwell, "Restorations of Masonic Geometry and Symbolry," I take this opportunity of presenting some
of the facts stressed by him in this connection.

The circle, though in itself intractable by any mathematical method as far as actual precision is
concerned nevertheless comes to the aid of mathematicians as the sole key to unlock the treasure house of
trigonometry and expose its exhaustless stores.

In all the processes in which the circle, or any part thereof, is directly involved, the ratio between the diameter and
the circumference is the fundamental truth to be first ascertained-for the diameter is what may be termed the
measure of the circle (i. e., the surface thereof). For it is always known or ascertainable by direct measurements.
And the circumference is that which must conform, according to some ascertained ratio, which, if correct, makes
all correct which it affects; but, if erroneous, infuses the error into all the results, and there it remains constantly
present.

As for instance, in the case of the velocity of the rim of a revolving wheel, or the length of an arc; but not in that
of the length of a radius, or of the spoke of a wheel, or the sine of an angle, for these last may be ascertained
without knowing the length of any circumference, or without any circle at all. Furthermore, all proportions between
the circumference and the parts of a circle, as chords, segments, sectors, etc., remain unaffected; and in calculating
the diameter of one circle that shall be of twice or thrice the surface of another of a certain diameter or
circumference, no harm can arise from an error in the ratio, for the operation only applies to the proportions
between certain parts of one figure, or the corresponding parts in two or more figures.

But when it is sought to ascertain the area or the length of the side of a square, hexagon, or other regular polygon,
or the several sides of an irregular figure, which shall be equal to or otherwise proportionately greater or less than
a circle of a given diameter, the error, if any, in the ratio between the diameter and the circumference, enters at
once into the work, and remains and propagates itself in every subsequent operation founded upon or involved with
it in most cases increasing as it proceeds. For, in finding the content or surface, the circumference must first be
known; and finding an equal square, triangle, or other figure, depends on first knowing the contents of the circle.
If the ratio depended on be too great or too small, the circumference, and consequently the area or surface therein
contained will be too great or too small. Hence in seeking a circle which shall contain a given surface say, equal
to a square of five on a side if the ratio should be too great, as 3.16, a diameter shorter than is correct must be
assigned to the circle, in order to bring the surface within the requirement of the ratio, that is, of the square.

The several methods of computing the surface of a circle of given diameter, depend on the ratio. One method is
to multiply the circumference by half the radius, or one-fourth the diameter, which is the same as to multiply the
diameter by one-fourth the circumference. No doubt a correct mode, if the ratio be the true one. Another method
is to multiply the square of the diameter (the square circumscribed about the circle) by one-fourth the ratio.

Still another method is to multiply the square of the radius by the ratio. By any one of these methods, the same
result is obtained, whether the ratio is right or wrong and the result will be right if the circumference be right,
i. e., if the ratio between the diameter and circumference, on which the latter is computed, be right; and as certainly
wrong if there be error in the ratio assumed for the purpose.

The conclusions reached by ninety-eight different authors, most of them skilled in mathematical pursuits, have
shown very positive but very different conclusions concerning this problem, already subjected to centuries of
continued dispute. The different methods of finding the ratio are not less than forty-four, and the different ratios
proposed, not less than seventy-two. The number in which the proportion between the diameter and the
circumference is greater than the commonly accepted ratio is about fifty-eight. The number giving a lower ratio than
the "orthodox" is sixteen, and the number of those which agree with the orthodox ratio is twenty-six. Of the whole
number, one hundred and seven bear date in this century. Now every specific numerical value assigned must be
wrong, except one; if any of them, by chance, should happen to be right.

The "orthodox" ratio depends for its validity on what Bromwell calls "the process of exhaustion," that is, in the
sufficiency and correctness of the work in the arithmetical computations of the area of a regular polygon having a
sufficient number of sides to render it substantially equivalent to a circle. No one of the modes of dealing with a
series of numbers or fractions, or known dimensions, of some part of a circle, or other figure, by multiplication,
division, etc., carries with itself its own demonstration. Had such been the case, there never would have been any
controversy. On the principle that several doubtful calculations make one good one, one of two solutions or both
are accepted because they agree, not because either is correct. Is it possible in any such case that any one can say,
before seeing the result, that the operation to be pursued is actually its own test and must be correct, or that it can
be referred to a veritable test? Some of the conclusions examined are manifestly false, while some others afford
nothing, except the assertions of the author, to show that the result is a ratio of anything.

If we take a polygon of four sides (a perfect square) and successively double the number of sides making it a
polygon of eight, of sixteen and then or thirty-two on so on, the sides will eventually become so numerous and so
short that the figure is so nearly a circle that the difference may be deemed of no consequence. However, at the
same time the content or surface of each polygon must be computed at every increase of the number of sides, by
means of two proportional triangles, involving multiplication, division, addition, extraction of the square root, etc.,
and the surface of the last polygon computed is accepted as the surface of the circle in question.

As every regular polygon (square, hexagon, polygon) may be considered as composed of as many isosceles triangles
(equal sided) as it has sides the side of the polygon being the base and the two equal sides of the triangle meeting
and forming an apex at the center of the polygon the circle may be regarded as a polygon of an indefinite number
of sides, and consequently composed of a like number of triangles, each having two equal and two very long sides
and an exceedingly short base, which is the same as the side of the polygon. Such a polygon may be regarded as
having a thousand million of equal sides, and consequently composed of as many equal sided triangles. If accurately
measured, such a polygon would doubtless furnish a very close approximation to the measure of a circle having a
radius equal to either of the two equal sides of any such triangle. But one with a less number of sides, say 30,000,
with a slight error in the computation of each triangle, might offer a grossly defective result. But it might be much
worse in case of a million sides, with an error in each.

As it appears from principles not dependent on any ratio between the diameter and circumference, the circumference
of a circle whose diameter is one is necessarily equal in figures to the area of a circle whose diameter is two, and
whoever succeeds in finding the surface of the latter circle, is thereby in possession of the number which shows the
true circumference of the former. Hence, the computation is made by ascertaining the area of a circle having a
diameter of two. To accomplish this, a polygon of four sides is circumscribed about the circle, and another of four
sides  the corresponding sides being made parallel is inscribed within it. The outer polygon is the same as the
square of the diameter, that is, its surface is equal to four, while the inscribed polygon (or square) is necessarily
one-half as much, equal to two; the side of the inscribed square being to the side of the circumscribed square as
the side of any square is to the line of its own diagonal so that the two squares (polygons) are in proportion to each
other as any two adjacent squares inscribed in any other circles for it can always be seen that the diameter of any
circle is the same as the diagonal line of its inscribed square. The object in taking two polygons is to secure a
convenient basis of measurement which would be lacking if only one were used. In doubling the number of sides
it necessarily comes to pass every time except in the case of the two polygons of four sides each that the angles
of the inscribed polygon present themselves to the middle of the sides of the exscribed polygon and vice versa. By
this a mean proportional as to surface, between corresponding parts of the two polygons, is also presented in
geometrical form, susceptible of being computed by ordinary mathematical rules. The beginning of this series of
duplications of the number of sides may be seen in the accompanying figure.

The forming and doubling the number of sides of these polygons is easy enough and the process may be continued
until the sides become so minute that no farther division is practicable, but however far carried it would go but little
toward finality, which is only to be reached by the computations. These give the measurements and demand the
utmost accuracy and here is where the trouble begins. The principal cause of difficulty is error, which attends the
work from first to last.

In order to reach the point at which the operator may intend to stop say at a polygon of 32,768 sides (the number
usually adopted) no less than twenty-seven complicated processes each made up of several partial or ancillary
operations must be accomplished. These are each simple enough, but they are not separate and independent, so
that any error, from omitted fractions or other causes, will only affect the particular calculation in which it may
occur, and there stop, but they are cumulative, as will be seen from what follows:

First, a simple expression of the surface content of a corresponding part of each of the two original polygons one
being 2, the other 4. Then a multiplication of these parts together and extraction of the square root of the product,
leaving a remainder. Then following twenty-seven operations, including thirty-six multiplications, involving sixty-
eight numbers, each containing an endless decimal fraction; also thirteen additions, each of two of the same
numbers, each with its fraction, being very nearly equal to, and slightly exceeding, so many multiplications by two,
the finding of thirteen quotients, and the extraction of thirteen additional square roots, each root and quotient leaving
a remainder. The entire process being one unbroken series of computations, every one dependent upon all which
precede it to the last. Any deficiency or excess in the first computation (which leaves a remainder) is thus multiplied
and remultiplied no less than sixty times, by a factor not less than 2.8 (and reaching 3.31; or more as an upper
limit, and thirteen times by two. And besides this we have the addition of two of the larger factors together thirteen
times, making it equivalent altogether to eighty-six multiplications by two in a series, each multiplying the former
product. And this all relates back to the first remainder, it being the first of twenty-six following in succession, each
on an average through forty-three multiplications, making more than one thousand one hundred multiplications in
all.

It is easy to form a square and a triangle equal in area to each other; and the same is true of any two figures
bounded by right lines however different their forms, for their lines are subject to direct and equal measurement,
and these being known, the included surfaces are easily dealt with. But not so with the circle. This remarkable
figure has something about it almost mysterious. While it is that by which all right lined figures may be proven as
to their forms, and in many cases even to their contents, yet to ascertain its own content, that is, to find an equal
square or other rightlined figure, has been a special object of search and the ever a present stumbling block of
mathematicians of all ages.

NOTE

It will be understood that the point raised in the article is theoretical rather than practical. All measurements are
approximate, and in the ease of the circle and other curved figures we have a second approximation. The first one
is the determination of the radius of the figure, and the second is that in the calculations to determine the ratio
between the radius or diameter and the circumference. And though, as has been shown, any error in these
calculations is a cumulative one, yet the process adopted puts a limit to the error. In subdividing the sides of the
inscribed and exscribed polygons shown in the second figure, it will be seen that the perimeter of the former
increases at every step, while the latter decreases. Eventually each becomes approximately coincident with the circle.
Thus every step in the calculations must fall between these limits and whatever error there may be cannot be great
enough to vitiate the result for any practical purpose. Ed.
