those of the Romans, but in certain directions even surpassed those of
the Greeks. In the most ancient source for the mathematics of India
that we know of, the Culvasûtras, which date back to a little before
our chronological era, we do not find, it is true, the squaring of
the circle treated of, but the opposite problem is dealt with, which
might fittingly be termed the circling of the square. The half of the
side of a given square is prolonged one third of the excess in length
of half the diagonal over half the side, and the line thus obtained
is taken as the radius of the circle equal in area to the square. The
simplest way to obtain an idea of the exactness of this construction
is to compute how great π would have to be if the construction were
exactly correct. We find out in this way that the value of π upon
which the Indian circling of the square is based, is about from five
to six hundredths smaller than the true value, whereas the approximate
π of Archimedes, 3-1/7, is only from one to two thousandths too large,
and the old Egyptian value exceeds the true value by from one to two
hundredths. Cyclometry very probably made great advances among the
Hindus in the first four or five centuries of our era; for Aryabhatta,
who lived about the year 500 after Christ, states, that the ratio of
the circumference to the diameter is 62832 : 20000, an approximation that
in exactness surpasses even that of Ptolemy. The Hindu result gives
3.1416 for π, while π really lies between 3.141592 and 3.141593. How
the Hindus obtained this excellent approximate value is told by Ganeça,
the commentator of Bhâskara, an author of the twelfth century. Ganeça
says that the method of Archimedes was carried still farther by the
Hindu mathematicians; that by continually doubling the number of sides
they proceeded from the hexagon to a polygon of 384 sides, and that by
the comparison of the circumferences of the inscribed and circumscribed
384-sided polygons they found that π was equal to 3927: 1250. It will
be seen that the value given by Bhâskara is identical with the value of
Aryabhatta. It is further worthy of remark that the earlier of these
two Hindu mathematicians does not mention either the value 3-1/7 of
Archimedes or the value 3-17/120 of Ptolemy, but that the later knows
of both values and especially recommends that of Archimedes as the
most useful one for practical application. Strange to say, the good
approximate value of Aryabhatta does not occur in Bramagupta, the great
Hindu mathematician who flourished in the beginning of the seventh
century; but we find the curious information in this author that the
area of a circle is exactly equal to the square root of 10 when the
radius is unity. The value of π as derivable from this formula,—a
value from two to three hundredths too large,—has unquestionably
arisen upon Hindu soil. For it occurs in no Grecian mathematician; and
Arabian authors, who were in a better position than we to know Greek
and Hindu mathematical literature, declare that the approximation
which makes π equal to the square root of 10, is of Hindu origin. It
is possible that the Hindu people, who were addicted more than any
other to numeral mysticism, sought to find in this approximation some
connection with the fact that man has ten fingers; and ten accordingly
is the basis of their numeral system.
Reviewing the achievements of the Hindus generally with respect to
the problem of the quadrature, we are brought to recognise that this
people, whose talents lay more in the line of arithmetical computation
than in the perception of spatial relations, accomplished as good as
nothing on the pure geometrical side of the problem, but that the merit
belongs to them of having carried the Archimedean method of computing π
several stages farther, and of having obtained in this way a much more
exact value for it—a circumstance that is explainable when we consider
that the Hindus are the inventors of our present system of numeral
notation, possessing which they easily outdid Archimedes, who employed
the awkward Greek system.
#Among the Chinese.#
With regard to the Chinese, this people operated in ancient times
with the Babylonian value for π, or 3; but possessed knowledge of
the approximate value of Archimedes at least since the end of the
sixth century. Besides this, there appears in a number of Chinese
mathematical treatises an approximate value peculiarly their own, in
which π = 3-7/50; a value, however, which notwithstanding it is written
in larger figures, is no better than that of Archimedes. Attempts at
the _constructive_ quadrature of the circle are not found among the
Chinese.
#Among the Arabs.#
Greater were the merits of the Arabians in the advancement and
development of mathematics; and especially in virtue of the fact that
they preserved from oblivion both Greek and Hindu mathematics, and
handed them down to the Christian countries of the West. The Arabians
expressly distinguished between the Archimedean approximate value and
the two Hindu values the square root of 10 and the ratio 62832 : 20000.
This distinction occurs also in Muhammed Ibn Musa Alchwarizmî, the
same scholar who in the beginning of the ninth century brought the
principles of our present system of numerical notation from India and
introduced the same into the Mohammedan world. The Arabians, however,
did not study the numerical quadrature of the circle only, but also
the constructive; as, for instance, Ibn Alhaitam, who lived in Egypt
about the year 1000 and whose treatise upon the squaring of the circle
is preserved in a Vatican codex, which has unfortunately not yet been
edited.
#In Christian times.#
Christian civilisation, to which we are now about to pass, produced up
to the second half of the fifteenth century extremely insignificant
results in mathematics. Even with regard to our present problem we have
but a single important work to mention; the work, namely, of Frankos
Von Lüttich, upon the squaring of the circle, published in six books,
but only preserved in fragments. The author, who lived in the first
half of the eleventh century, was probably a pupil of Pope Sylvester
II, himself a not inconsiderable mathematician for his time, and who
also wrote the most celebrated book on geometry of the period.
#Cardinal Nicolaus De Cusa.#
Greater interest came to be bestowed upon mathematics in general, but
especially on the problem of the quadrature of the circle, in the
second half of the fifteenth century, when the sciences again began
to revive. This interest was especially aroused by Cardinal Nicolaus
De Cusa, a man highly esteemed on account of his astronomical and
calendarial studies. He claimed to have discovered the quadrature
of the circle by the employment solely of compasses and ruler, and
thus attracted the attention of scholars to the now historic problem.
People believed the famous Cardinal, and marvelled at his wisdom,
until Regiomontanus, in letters which he wrote in 1464 and 1465 and
which were published in 1533, rigidly demonstrated that the Cardinal's
quadrature was incorrect. The construction of Cusa was as follows. The
radius of a circle is prolonged a distance equal to the side of the
inscribed square; the line thus obtained is taken as the diameter of a
second circle and in the latter an equilateral triangle is described;
then the perimeter of the latter is equal to the circumference of the
original circle. If this construction, which its inventor regarded as
exact, be considered as a construction of approximation, it will be
found to be more inexact even than the construction resulting from
the value π = 3-1/7. For by Cusa's method π would be from five to six
thousandths smaller than it really is.
#Bovillius and Orontius Finaeus.#
In the beginning of the sixteenth century a certain Bovillius appears,
who announced anew the construction of Cusa; meeting however with
no notice. But about the middle of the sixteenth century a book was
published which the scholars of the time at first received with
interest. It bore the proud title "De Rebus Mathematicis Hactenus
Desideratis." Its author, Orontius Finaeus, represented that he had
overcome all the difficulties that had ever stood in the way of
geometrical investigators; and incidentally he also communicated to the
world the "true quadrature" of the circle. His fame was short-lived.
For soon afterwards, in a book entitled "De Erratis Orontii," the
Portuguese Petrus Nonius demonstrated that Orontius's quadrature, like
most of his other professed discoveries, was incorrect.
#Simon Van Eyck.#
In the period following this the number of circle-squarers so increased
that we shall have to limit ourselves to those whom mathematicians
recognise. And particularly is Simon Van Eyck to be mentioned, who
towards the close of the sixteenth century published a quadrature which
was so approximate that the value of π derived from it was more exact
than that of Archimedes; and to disprove it the mathematician Peter
Metius was obliged to seek a still more accurate value than 3-1/7. The
erroneous quadrature of Van Eyck was thus the occasion of Metius's
discovery that the ratio 355 : 113, or 3-16/113, varied from the true
value of π by less than one one-millionth, eclipsing accordingly all
values hitherto obtained. Moreover, it is demonstrable by the theory of
continued fractions, that, admitting figures to four places only, no
two numbers more exactly represent the value of π than 355 and 113.
#Joseph Scaliger.#
In the same way the quadrature of the great philologist Joseph
Scaliger led to refutations. Like most circle-squarers who believe
in their discovery, Scaliger also was little versed in the elements
of geometry. He solved, however,—at least in his own opinion he
did,—the famous problem; and published in 1592 a book upon it, which
bore the pretentious title "Nova Cyclometria" and in which the name of
Archimedes was derided. The worthlessness of his supposed discovery was
demonstrated to him by the greatest mathematicians of his time; namely,
Vieta, Adrianus Romanus, and Clavius.
#Longomontanus, John Porta, and Gregory St. Vincent.#
Of the erring circle-squarers that flourished before the middle
of the seventeenth century three others deserve particular
mention—Longomontanus of Copenhagen, who rendered such great services
to astronomy, the Neapolitan John Porta, and Gregory of St. Vincent.
Longomontanus made π = 3-14185/100000, and was so convinced of the
correctness of his result that he thanked God fervently, in the preface
to his work "Inventio Quadraturae Circuli," that He had granted him in
his high old age the strength to conquer the celebrated difficulty.
John Porta followed the initiative of Hippocrates, and believed he had
solved the problem by the comparison of lunes. Gregory of St. Vincent
published a quadrature, the error of which was very hard to detect but
was finally discovered by Descartes.
#Peter Metius and Vieta.#
Of the famous mathematicians who dealt with our problem in the period
between the close of the fifteenth century and the time of Newton,
we first meet with Peter Metius, before mentioned, who succeeded
in finding in the fraction 355 : 113 the best approximate value for
π involving only small numbers. The problem received a different
advancement at the hands of the famous mathematician Vieta. Vieta
was the first to whom the idea occurred of representing π with
mathematical exactness by an infinite series of continuable operations.
By comparison of inscribed and circumscribed polygons, Vieta found
that we approach nearer and nearer to π if we allow the operations
of the extraction of the square root of 1/2, and of addition and of
multiplication to succeed each other in a certain manner, and that
π must come out exactly, if this series of operations could be
indefinitely continued. Vieta thus found that to a diameter of 10000
million units a circumference belongs of 31415 million and from 926535
to 926536 units of the same length.
#Adrianus Romanus, Ludolf Van Ceulen.#
But Vieta was outdone by the Netherlander Adrianus Romanus, who added
five additional decimal places to the ten of Vieta. To accomplish this
he computed with unspeakable labor the circumference of a regular
circumscribed polygon of 1073741824 sides. This number is the thirtieth
power of 2. Yet great as the labor of Adrianus Romanus was, that
of Ludolf Van Ceulen was still greater; for the latter calculator
succeeded in carrying the Archimedean process of approximation for the
value of π to 35 decimal places, that is, the deviation from the true
value was smaller than one one-thousand quintillionth, a degree of
exactness that we can hardly have any conception of. Ludolf published
the figures of the tremendous computation that led to this result. His
calculation was carefully examined by the mathematician Griemberger
and declared to be correct. Ludolf was justly proud of his work, and
following the example of Archimedes, requested in his will that the
result of his most important mathematical performance, the computation
of π to 35 decimal places, be engraved upon his tombstone; a request
which is said to have been carried out. In honor of Ludolf, π is called
to-day in Germany the Ludolfian number.
#The new method of Snell. Huygens's verification of it.#
Although through the labor of Ludolf a degree of exactness for
cyclometrical operations was now obtained that was more than sufficient
for any practical purpose that could ever arise, neither the problem
of constructive rectification nor that of constructive quadrature was
thereby in any respect theoretically advanced. The investigations
conducted by the famous mathematicians and physicists Huygens and
Snell about the middle of the seventeenth century, were more important
from a mathematical point of view than the work of Ludolf. In his book
"Cyclometricus" Snell took the position that the method of comparison
of polygons, which originated with Archimedes and was employed by
Ludolf, need by no means be the best method of attaining the end
sought; and he succeeded by the employment of propositions which state
that certain arcs of a circle are greater or smaller than certain
straight lines connected with the circle, in obtaining methods that
make it possible to reach results like the Ludolfian with much less
labor of calculation. The beautiful theorems of Snell were proved a
second time, and better proved, by the celebrated Dutch promoter of the
science of optics, Huygens (Opera Varia, p. 365 et seq.; "Theoremata De
Circuli et Hyperbolae Quadratura," 1651), as well as perfected in many
ways. Snell and Huygens were fully aware that they had advanced only
the problem of numerical quadrature, and not that of the constructive
quadrature. This, in Huygens's case, plainly appeared from the vehement
dispute he conducted with the English mathematician James Gregory.
This controversy has some significance for the history of our problem,
from the fact that Gregory made the first attempt to prove that the
squaring of the circle with ruler and compasses must be impossible.
#The controversy between Huygens and Gregory.# The result of the
controversy, to which we owe many valuable treatises, was, that Huygens
finally demonstrated in an incontrovertible manner the incorrectness
of Gregory's proof of impossibility, adding that he also was of
opinion that the solution of the problem with ruler and compasses was
impossible, but nevertheless was not himself able to demonstrate this
fact. And Newton, later, expressed himself to a similar effect. As a
matter of fact it took till the most recent period, that is over 200
years, until higher mathematics was far enough advanced to furnish a
rigid demonstration of impossibility.
V.
Before we proceed to consider the promotive influence which the
invention of the differential and the integral calculus had upon our
problem, we shall enumerate a few at least of that never-ending line
of mistaken quadrators who delighted the world by the fruits of their
ingenuity from the time of Newton to the present period; and out of a
pious and sincere consideration for the contemporary world, we shall
entirely omit in this to speak of the circle-squarers of our own time.
#Hobbes's quadrature.#
First to be mentioned is the celebrated English philosopher Hobbes.
In his book "De Problematis Physicis," in which he chiefly proposes
to explain the phenomena of gravity and of ocean tides, he also takes
up the quadrature of the circle and gives a very trivial construction
that in his opinion definitively solved the problem, making π = 3-1/5.
In view of Hobbes's importance as a philosopher, two mathematicians,
Huygens and Wallis, thought it proper to refute Hobbes at length.
But Hobbes defended his position in a special treatise, in which
to sustain at least the appearance of being right, he disputed the
fundamental principles of geometry and the theorem of Pythagoras; so
that mathematicians could pass on from him to the order of the day.
#French quadrators of the Eighteenth Century.#
In the last century France especially was rich in circle-squarers.
We will mention: Oliver de Serres, who by means of a pair of scales
determined that a circle weighed as much as the square upon the side
of the equilateral triangle inscribed in it, that therefore they
must have the same area, an experiment in which π = 3; Mathulon, who
offered in legal form a reward of a thousand dollars to the person
who would point out an error in his solution of the problem, and who
was actually compelled by the courts to pay the money; Basselin, who
believed that his quadrature must be right because it agreed with the
approximate value of Archimedes, and who anathematised his ungrateful
contemporaries, in the confidence that he would be recognised by
posterity; Liger, who proved that a part is greater than the whole
and to whom therefore the quadrature of the circle was child's play;
Clerget, who based his solution upon the principle that a circle is a
polygon of a definite number of sides, and who calculated, also, among
other things, how large the point is at which two circles touch.
#Germany and Poland.#
Germany and Poland also furnish their contingent to the army of
circle-squarers. Lieutenant-Colonel Corsonich produced a quadrature
in which π equalled 3-1/8, and promised fifty ducats to the person
who could prove that it was incorrect. Hesse of Berlin wrote an
arithmetic in 1776, in which a true quadrature was also "made known,"
π being exactly equal to 3-14/99. About the same time Professor
Bischoff of Stettin defended a quadrature previously published by
Captain Leistner, Preacher Merkel, and Schoolmaster Böhm, which made
π _implicite_ equal to the square of 62/35, not even attaining the
approximation of Archimedes.
#Constructive approximations. Euler. Kochansky.#
From attempts of this character are to be clearly distinguished
constructions of approximation in which the inventor is aware that
he has not found a mathematically exact construction, but only
an approximate one. The value of such a construction will depend
upon two things—first, upon the degree of exactness with which
it is numerically expressed, and secondly on the fact whether the
construction can be more or less easily made with ruler and compasses.
Constructions of this kind, simple in form and yet sufficiently exact
for practical purposes, have for centuries been furnished us in great
numbers. The great mathematician Euler, who died in 1783, did not think
it out of place to attempt an approximate construction of this kind. A
very simple construction for the rectification of the circle and one
which has passed into many geometrical text books, is that published
by Kochansky in 1685 in the _Leipziger Berichte_. It is as follows:
"Erect upon the diameter of a circle at its extremities perpendiculars;
with the centre as vertex, mark off upon the diameter an angle of 30°;
find the point of intersection with the perpendicular of the line
last drawn, and join this point of intersection with that point upon
the other perpendicular which is at a distance of three radii from
the base of the perpendicular. The line of junction thus obtained is
then very approximately equal to one-half of the circumference of the
given circle." Calculation shows that the difference between the true
length of the circumference and the line thus constructed is less than
3/100000 of the diameter.
#Inutility of constructive approximations.#
Although such constructions of approximation are very interesting
in themselves, they nevertheless play but a subordinate rôle in the
history of the squaring of the circle; for on the one hand they can
never furnish greater exactness for circle-computation than the
thirty-five decimal places which Ludolf found, and on the other hand
they are not adapted to advance in any way the question whether the
exact quadrature of the circle with ruler and compasses is possible.
#The researches of Newton, Leibnitz, Wallis, and Brouncker.#
The numerical side of the problem, however, was considerably advanced
by the new mathematical methods perfected by Newton and Leibnitz,
commonly called the differential and the integral calculus. And about
the middle of the seventeenth century, some time before Newton and
Leibnitz represented π by series of powers, the English mathematicians
Wallis and Lord Brouncker, Newton's predecessors in a certain sense,
succeeded in representing π by an infinite series of figures combined
by the first four rules of arithmetic. A new method of computation was
thus opened. Wallis found that the fourth part of π is represented more
exactly by the regularly formed product
2/3 × 4/3 × 4/5 × 6/5 × 6/7 × 8/7 × 8/9 × etc.
the farther the multiplication is continued, and that the result always
comes out too small if we stop at a proper fraction but too large if
we stop at an improper fraction. Lord Brouncker, on the other hand,
represents the value in question by a continued fraction in which all
the denominators are equal to 2 and the numerators are odd square
numbers. Wallis, to whom Brouncker had communicated his elegant result
without proof, demonstrated the same in his "Arithmetic of Infinites."
The computation of π could hardly be farther advanced by these results
than Ludolf and others had carried it, though of course in a more
laborious way. However, the series of powers derived by the assistance
of the differential calculus of Newton and Leibnitz furnished a means
of computing it to hundreds of decimal places.
#Other calculations.#
Gregory, Newton, and Leibnitz next found that the fourth part of π was
equal exactly to
1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ...
if we conceive this series, which is called the Leibnitzian,
indefinitely continued. This series is indeed wonderfully simple, but
is not adapted to the computation of π, for the reason that entirely
too many members have to be taken into account to obtain π accurately
to a few decimal places only. The original formula, however, from which
this series is derived, gives other formulas which are excellently
adapted to the actual computation. This formula is the general series:
α = _a_ - 1/3_a_^3 + 1/5_a_^5 - 1/7_a_^7 + ...,
where α is the length of the arc that belongs to any central angle in
a circle of radius 1, and where _a_ is the tangent to this angle. From
this we derive the following:
π/4 = (_a_ + _b_ + _c_ + ...) - 1/3(_a_^3 + _b_^3 + _c_^3 + ...)
+ 1/5(_a_^5 + _b_^5 + _c_^5 + ...) - ...,
where _a_, _b_, _c_ ... are the tangents of angles whose sum is
45°. Determining, therefore, the values of _a_, _b_, _c_ ..., which
are equal to small and easy fractions and fulfil the condition just
mentioned, we obtain series of powers which are adapted to the
computation of π. The first to add by the aid of series of this
description additional decimal places to the old 35 in the number π
was the English arithmetician Abraham Sharp, who following Halley's
instructions, in 1700, worked out π to 72 decimal places. A little
later Machin, professor of astronomy in London, computed π to 100
decimal places; putting, in the series given above, _a_ = _b_ = _c_ =
_d_ = 1/5 and _e_ =-1/239, that is employing the following series:
π/4 = 4. [1/5 - 1/3.5^3 + 1/5.5^5 - 1/7.5^7 + ...]
- [1/239 - 1/3.239^3 + 1/5.239^5 - ...]
#The computation of π to many decimal places.#
In the year 1819, Lagny of Paris outdid the computation of Machin,
determining in two different ways the first 127 decimal places of π.
Vega then obtained as many as 140 places, and the Hamburg arithmetician
Zacharias Dase went as far as 200 places. The latter did not use
Machin's series in his calculation, but the series produced by putting
in the general series above given _a_ = 1/2, _b_ = 1/5, _c_ = 1/8.
Finally, at a recent date, π has been computed to 500 places.
#Idea of exactness obtainable with the approximate values of π.#
The computation to so many decimal places may serve as an illustration
of the excellence of the modern method as contrasted with those
anciently employed, but otherwise it has neither a theoretical nor a
practical value. That the computation of π to say 15 decimal places
more than sufficiently satisfies the subtlest requirements of practice
may be gathered from a concrete example of the degree of exactness thus
obtainable. Imagine a circle to be described with Berlin as centre, and
the circumference to pass through Hamburg; then let the circumference
of the circle be computed by multiplying its diameter with the value of
π to 15 decimal places, and then conceive it to be actually measured.
The deviation from the true length in so large a circle as this even
could not be as great as the 18 millionth part of a millimetre.