number π is not equal to the ratio of two whole numbers however great,
it does not immediately follow that it is impossible to construct
a straight line exactly equal in length to the circumference of a
circle; thus demonstrating at once the impossibility of solving the
problem. This question is to be answered in the negative. For there
are in geometry many sets of two lines of which the one can be easily
constructed from the other, notwithstanding the fact that no two whole
numbers can be found to represent the ratio of the two lines. The side
and the diagonal of a square, for instance, are so constituted. It is
true the ratio of the latter two magnitudes is nearly that of 5 to 7.
But this proportion is not exact, and there are in fact no two numbers
that represent the ratio exactly. Nevertheless, either of these two
lines can be easily constructed from the other by the sole employment
of ruler and compasses. This might be the case, too, with the
rectification of the circle; and consequently from the impossibility of
representing π by the ratio between two whole numbers the impossibility
of the problem of rectification is not inferable.
The quadrature of the circle stands and falls with the problem of
rectification. This is based upon the truth above mentioned, that
a circle is equal in area to a right-angled triangle, in which one
side is equal to the radius of the circle and the other to the
circumference. Supposing, accordingly, that the circumference of the
circle were rectified, then we could construct this triangle. But every
triangle, as is taught in the elements of planimetry, can, with the
help of ruler and compasses be converted into a square exactly equal
to it in area. So that, therefore, supposing the rectification of the
circumference of a circle were successfully performed, a square could
be constructed that would be exactly equal in area to the circle.
The dependence upon one another of the three problems of the
computation of the number π, of the quadrature of the circle, and its
rectification, thus obliges us, in dealing with the history of the
quadrature, to regard investigations with respect to the value of π and
attempts to rectify the circle as of equal importance, and to consider
them accordingly.
#Conditions of the geometrical solution.#
We have used repeatedly in the course of this discussion the expression
"to construct with ruler and compasses." It will be necessary to
explain what is meant by the specification of these two instruments.
When such a number of conditions is annexed to a requirement in
geometry to construct a certain figure that the construction only of
_one_ figure or a limited number of figures is possible in accordance
with the conditions given; such a complete requirement is called a
problem of construction, or briefly a problem. When a problem of this
kind is presented for solution it is necessary to reduce it to simpler
problems, already recognised as solvable; and since these latter depend
in their turn upon other, still simpler problems, we are finally
brought back to certain fundamental problems upon which the rest are
based but which are not themselves reducible to problems less simple.
These fundamental problems are, so to speak, the undermost stones of
the edifice of geometrical construction. The question next arises as to
what problems may be properly regarded as fundamental; and it has been
found, that the solution of a great part of the problems that arise in
elementary planimetry rests upon the solution of only five original
problems. They are:
1. The construction of a straight line which shall pass through two
given points.
2. The construction of a circle the centre of which is a given point
and the radius of which has a given length.
3. The determination of the point that lies coincidently on two given
straight lines extended as far as is necessary,—in case such a point
(point of intersection) exists.
4. The determination of the two points that lie coincidently on a given
straight line and a given circle,—in case such common points (points
of intersection) exist.
5. The determination of the two points that lie coincidently on two
given circles,—in case such common points (points of intersection)
exist.
For the solution of the three last of these five problems the eye
alone is needed, while for the solution of the two first problems,
besides pencil, ink, chalk, and the like, additional special
instruments are required: for the solution of the first problem a
ruler is most generally used, and for the solution of the second a
pair of compasses. But it must be remembered that it is no concern
of geometry what mechanical instruments are employed in the solution
of the five problems mentioned. Geometry simply limits itself to
the presupposition that these problems are solvable, and regards
a complicated problem as solved if, upon a specification of the
constructions of which the solution consists, no other requirements are
demanded than the five above mentioned. Since, accordingly, geometry
does not itself furnish the solution of these five problems, but
rather exacts them, they are termed _postulates_.[51] All problems of
planimetry are not reducible to these five problems alone. There are
problems that can be solved only by assuming other problems as solvable
which are not included in the five given; for example, the construction
of an ellipse, having given its centre and its major and minor axes.
Many problems, however, possess the property of being solvable with the
assistance solely of the five postulates above formulated, and where
this is the case they are said to be "constructible with ruler and
compasses," or "elementarily" constructible.
[51] Usually geometers mention only two postulates (Nos. 1 and
2). But since to geometry proper it is indifferent whether
only the eye, or additional special mechanical instruments are
necessary, the author has regarded it more correct in point of
method to assume five postulates.
After these general remarks upon the solvability of problems of
geometrical construction, which an understanding of the history of the
squaring of the circle makes indispensably necessary, the significance
of the question whether the quadrature of the circle is or is not
solvable, that is elementarily solvable, will become intelligible.
But the conception just discussed of elementary solvability only
gradually took clear form, and we therefore find among the Greeks as
well as among the Arabs, endeavors, successful in some respects, that
aimed at solving the quadrature of the circle with other expedients
than the five postulates. We have also to take these endeavors into
consideration, and especially so as they, no less than the unsuccessful
efforts at elementary solution, have upon the whole advanced the
science of geometry, and contributed much to the clarification of
geometrical ideas.
III.
#The Egyptian quadrature.#
In the oldest mathematical work that we possess we find a rule that
tells us how to make a square which is equal in area to a given
circle. This celebrated book, the Papyrus Rhind of the British Museum,
translated and explained by Eisenlohr (Leipsic, 1887), was written,
as it is stated in the work, in the thirty-third year of the reign of
King Ra-a-us, by a scribe of that monarch, named Ahmes. The composition
of the work falls accordingly into the period of the two Hiksos
dynasties, that is, in the period between 2000 and 1700 B.C. But there
is another important circumstance attached to this. Ahmes mentions
in his introduction that he composed his work after the model of old
treatises, written in the time of King Raenmat; whence it appears that
the originals of the mathematical expositions of Ahmes, are half a
thousand years older yet than the Papyrus Rhind.
The rule given in this papyrus for obtaining a square equal to a
circle, specifies that the diameter of the circle shall be shortened
one ninth of its length and upon the shortened line thus obtained a
square erected. Of course, the area of a square of this construction
is only approximately equal to the area of the circle. An idea may
be obtained of the degree of exactness of this original, primitive
quadrature by our remarking, that if the diameter of the circle in
question is one metre in length, the square that is supposed to be
equal to the circle is a little less than half a square decimetre
larger; an approximation not so accurate as that computed by
Archimedes, yet much more correct than many a one later employed. It
is not known how Ahmes or his predecessors arrived at this approximate
quadrature; but it is certain that it was handed down in Egypt from
century to century, and in late Egyptian times it repeatedly appears.
#The Biblical and Babylonian quadratures.#
Besides among the Egyptians, we also find in pre-Grecian antiquity an
attempt at circle-computation among the Babylonians. This is not a
quadrature; but aims at the rectification of the circumference. The
Babylonian mathematicians had discovered, that if the radius of a
circle be successively inscribed as chord within its circumference,
after the sixth inscription we arrive at the point of departure, and
they concluded from this that the circumference of a circle must be a
little larger than a line which is six times as long as the radius,
that is three times as long as the diameter. A trace of this Babylonian
method of computation may even be found in the bible; for in 1 Kings
vii. 23, and 2 Chron. iv. 2, the great laver is described, which under
the name of the "molten sea" constituted an ornament of the temple of
Solomon; and it is said of this vessel that it measured ten cubits from
brim to brim, and thirty cubits round about. The number 3 as the ratio
between the circumference and the diameter is still more plainly given
in the Talmud, where we read that "that which measures three lengths in
circumference is one length across."
#Among the Greeks.#
With regard to the earlier Greek mathematicians,—as Thales and
Pythagoras,—we know that they acquired the foundations of their
mathematical knowledge in Egypt. But nothing has been handed down
to us which shows that they knew of the old Egyptian quadrature, or
that they dealt with the problem at all. But tradition says, that,
subsequently, the teacher of Euripides and Pericles, the great
philosopher and mathematician Anaxagoras, whom Plato so highly praised,
"drew the quadrature of the circle" in prison, in the year 434. This
is the account of Plutarch in the seventeenth chapter of his work "De
Exilio." #Anaxagoras.# The method is not told us in which Anaxagoras
had supposably solved the problem, and it is not said whether knowingly
or unknowingly he accomplished an approximate solution after the manner
of Ahmes. But at any rate, to Anaxagoras belongs the merit of having
called attention to a problem that bore great fruit, in having incited
Grecian scholars to busy themselves with geometry, and thus more and
more to advance that science.
#The quadratrix of Hippias of Elis.#
Again, it is reported that the mathematician Hippias of Elis invented
a curved line that could be made to serve a double purpose: first,
to trisect an angle, and second, to square the circle. This curved
line is the τετραγωνίστουσα so often mentioned by the later Greek
mathematicians, and by the Romans called "quadratrix." Regarding the
nature of this curve we have exact knowledge from Pappus. But it will
be sufficient, here, to state that the quadratrix is not a circle
nor a portion of a circle, so that its construction is not possible
by means of the postulates enumerated in the preceding section. And
therefore the solution of the quadrature of the circle founded on the
construction of the quadratrix is not an elementary solution in the
sense discussed in the last section. We can, it is true, conceive a
mechanism that will draw this curve as well as compasses draw a circle;
and with the assistance of a mechanism of this description the squaring
of the circle is solvable with exactitude. But if it be allowed to
employ in a solution an apparatus especially adapted thereto, every
problem may be said to be solvable. Strictly taken, the invention
of the curve of Hippias substitutes for one insuperable difficulty
another equally insuperable. Some time afterwards, about the year 350,
the mathematician Dinostratus showed that the quadratrix could also
be used to solve the problem of rectification, and from that time on
this problem plays almost the same rôle in Grecian mathematics as the
related problem of quadrature.
#The Sophists' solution.#
As these problems gradually became known to the non-mathematicians of
Greece, attempts at solution at once sprang up that are worthy of a
place by the side of the solutions of modern amateur circle-squarers.
The Sophists, especially, believed themselves competent by seductive
dialectic to take a stronghold that had defied the intellectual
onslaughts of the greatest mathematicians. With verbal nicety,
amounting to puerility, it was said that the squaring of the circle
depended upon the finding of a number which represented in itself
both a square and a circle; a square by being a square number, a
circle in that it ended with the same number as the root number from
which, by multiplication with itself, it was produced. The number 36,
accordingly, was, as they thought, the one that embodied the solution
of the famous problem.
#Antiphon's attempt.#
Contrasted with this twisting of words the speculations of Bryson and
Antiphon, both contemporaries of Socrates, though inexact, appear in
high degree intelligent. Antiphon divided the circle into four equal
arcs, and by joining the points of division obtained a square; he
then divided each arc again into two equal parts and thus obtained an
inscribed octagon; thence he constructed an inscribed dodecagon, and
perceived that the figure so inscribed more and more approached the
shape of a circle. In this way, he said, one should proceed, until
there was inscribed in the circle a polygon whose sides by reason of
their smallness should coincide with the circle. Now this polygon
could, by methods already taught by the Pythagoreans, be converted
into a square of equal area; and upon the basis of this fact Antiphon
regarded the squaring of the circle as solved.
Nothing can be said against this method except that, however far the
bisection of the arcs is carried, the result must still remain an
approximate one.
#Bryson of Heraclea.#
The attempt of Bryson of Heraclea was better still; for this scholar
did not rest content with finding a square that was very little smaller
than the circle, but obtained by means of circumscribed polygons
another square that was very little larger than the circle. Only Bryson
committed the error of believing that the area of the circle was the
arithmetical mean between an inscribed and a circumscribed polygon
of an equal number of sides. Notwithstanding this error, however, to
Bryson belongs the merit, first, of having introduced into mathematics
by his emphasis of the necessity of a square which was too large and
one which was too small, the conception of maximum and minimum "limits"
in approximations; and secondly, by his comparison with a circle of
the inscribed and circumscribed regular polygons, the merit of having
indicated to Archimedes the way by which an approximate value for π was
to be reached.
#Hippocrates of Chios.#
Not long after Antiphon and Bryson, Hippocrates of Chios treated the
problem, which had now become more and more famous, from a new point
of view. Hippocrates was not satisfied with approximate equalities,
and searched for curvilinearly bounded plane figures which should be
mathematically equal to a rectilinearly bounded figure, and therefore
could be converted by ruler and compasses into a square equal in area.
First, Hippocrates found that the crescent-shaped plane figure produced
by drawing two perpendicular radii in a circle and describing upon the
line joining their extremities a semicircle, is exactly equal in area
to the triangle that is formed by this line of junction and the two
radii; and upon the basis of this fact the endeavors of the untiring
scholar were directed towards converting a circle into a crescent.
Naturally he was unable to attain this object, but by his efforts to
this end he discovered many a new geometrical truth; among others the
generalised form of the theorem mentioned, which bears to the present
day the name of "Lunulae Hippocratis," the lunes of Hippocrates. Thus
it appears, in the case of Hippocrates, in the plainest light, how the
very insolvable problems of science are qualified to advance science;
in that they incite investigators to devote themselves with persistence
to its study and thus to fathom its depths.
#Euclid's avoidance of the problem.#
Following Hippocrates in the historical line of the great Grecian
geometricians comes the systematist Euclid, whose rigid formulation of
geometrical principles has remained the standard presentation down to
the present century. The Elements of Euclid, however, contain nothing
relating to the quadrature of the circle or to circle-computation.
Comparisons of surfaces which relate to the circle are indeed found in
the book, but nowhere a computation of the circumference of a circle
or of the area of a circle. This palpable gap in Euclid's system was
filled by Archimedes, the greatest mathematician of antiquity.
#Archimedes's calculations.#
Archimedes was born in Syracuse in the year 287 B. C., and devoted his
life, there spent, to the mathematical and the physical sciences, which
he enriched with invaluable contributions. He lived in Syracuse till
the taking of the town by Marcellus, in the year 212 B. C., when he
fell by the hand of a Roman soldier whom he had forbidden to destroy
the figures he had drawn in the sand. To the greatest performances of
Archimedes the successful computation of the number π unquestionably
belongs. Like Bryson he started with regular inscribed and
circumscribed polygons. He showed how it was possible, beginning with
the perimeter of an inscribed hexagon, which is equal to six radii,
to obtain by way of calculation the perimeter of a regular dodecagon,
and then the perimeter of a figure having double the number of sides
of the preceding one. Treating, then, the circumscribed polygons in a
similar manner, and proceeding with both series of polygons up to a
regular 96-sided polygon, he perceived on the one hand that the ratio
of the perimeter of the inscribed 96-sided polygon to the diameter
was greater than 6336 : 2017-1/4, and on the other hand, that the
corresponding ratio with respect to the circumscribed 96-sided polygon
was smaller than 14688 : 4673-1/2. He inferred from this, that the number
π, the ratio of the circumference to the diameter, was greater than
the fraction 6336/2017-1/4 and smaller than 14688/4673-1/2. Reducing
the two limits thus found for the value of π, Archimedes then showed
that the first fraction was greater than and that 3-10/71 and that
the second fraction was smaller than 3-1/7, whence it followed with
certainty that the value sought for π lay between 3-1/7 and 3-10/71.
The larger of these two approximate values is the only one usually
learned and employed. That which fills us most with astonishment in the
Archimedean computation of π, is, first, the great acumen and accuracy
displayed in all the details of the computation, and then the unwearied
perseverance that he must have exercised in calculating the limits of
π without the advantages of the Arabian system of numerals and of the
decimal notation. For it must be considered that at many stages of the
computation what we call the extraction of roots was necessary, and
that Archimedes could only by extremely tedious calculations obtain
ratios that expressed approximately the roots of given numbers and
fractions.
#The later mathematicians of Greece.#
With regard to the mathematicians of Greece that follow Archimedes,
all refer to and employ the approximate value of 3-1/7 for π, without
however, contributing anything essentially new or additional to the
problems of quadrature and of cyclometry. Thus Heron of Alexandria,
the father of surveying, who flourished about the year 100 B. C.,
employs for purposes of practical measurement sometimes the value
3-1/7 for π and sometimes even the rougher approximation π = 3. The
astronomer Ptolemy, who lived in Alexandria about the year 150 A.
D., and who was famous as being the author of the planetary system
universally recognised as correct down to the time of Copernicus, was
the only one who furnished a more exact value; this he designated, in
the sexigesimal system of fractional notation which he employed, by 3,
8, 30,—that is 3 and 8/60 and 30/3600, or as we now say 3 degrees,
8 minutes (partes minutae primae), and 30 seconds (partes minutae
secundae). As a matter of fact, the expression 3 + 8/60 + 30/3600 =
3-17/120 represents the number π more exactly than 3-1/7; but on the
other hand, is, by reason of the magnitude of the numbers 17 and 120 as
compared with the numbers 1 and 7, more cumbersome.
IV.
#Among the Romans.#
In the mathematical sciences, more than in any other, the Romans stood
upon the shoulders of the Greeks. Indeed, with respect to cyclometry,
they not only did not add anything to the Grecian discoveries, but
often evinced even that they either did not know of the beautiful
result obtained by Archimedes, or at least did not know how to
appreciate it. For instance, Vitruvius, who lived during the time
of Augustus, computed that a wheel 4 feet in diameter must measure
12-1/2 feet in circumference; in other words, he made π equal to
3-1/8. And, similarly, a treatise on surveying, preserved to us in
the Gudian manuscript of the library at Wolfenbüttel, contains the
following instructions to square the circle: Divide the circumference
of a circle into four parts and make one part the side of a square;
this square will be equal in area to the circle. Aside from the fact
that the rectification of the arc of a circle is requisite to the
construction of a square of this kind, the Roman quadrature, viewed as
a calculation, is more inexact even than any other computation; for its
result is that π = 4.
#Among the Hindus.#