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Title: The lemniscate and Fagnano's contributions to elliptic integrals

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The Lemniscate and Fagnano's Contributions
to Elliptic Integrals

Communicated by C. TRUESDELL

§ 1. Introduction
C. G. J. JACOBI (1804-1851) called December 23, 1751 the brithday of elliptic
functions. It was on this date that L. EULER (1707-1783) was asked to examine
the collected papers of Count GIULIO FAGNANO (1682-1766) [1], who was being
proposed for membership in the Berlin Academy.
Many of the papers in FAGNANO'S works dealt with integrals which are not
expressible by finitely many elementary functions yet have a geometric interpretation. These were related to one another and, while interesting in themselves,
were not, in general, noteworthy. One of these, however, struck EULER forcefully.
It is called "Method for measuring the lemniscate" and was published in 1718.
What had EULER seen that excited him and that prompted the comment of JACOBI .9

FAGNANO'S paper is in three parts and deals, among other things, with the
division of the arc of a lemniscate. This requires a simple case of what is now
called complex multiplication. The paper begins by referring to a paper [2] of
JOHN BERNOULLI (1667-1748) to justify his investigations. There is a certain ad
hoc character to the results and although C. L. SIEGEL [3] has made an effort
to clarify the motivation, the origins of FAGNANO'S transformations remain obscure.
FAGNANO'S paper studies the function which has come to be called the lemniscate integral, viz.





It should be noted however, that the integral had arisen some 30 years earlier
in connection with a special kind of elastic curve called the "rectangular elastica"
and was studied by JAMES BERNOULLI about 1691 though BERNOULLI did not
publish details until 1694 [4]. Thus, while the designation of the integral as the
lemniscate integral is now common, its appellation as the "curva elastica" has



strong historical justification. Indeed in his diary, entry 51, GAuss starts to write
"Curvam elasticam" but crosses out "elasticam" and replaces it with "lemniscatam".
EULER was very much interested in elasticity and in the integral given above,
and he wrote a paper in 1728 dealing with these questions 1 [5]. I shall refer again
to the work of EOLER and FAGNANO in § 3. Readers will find the account of
C. TRUESDELL [6] both highly interesting and thorough, with references to the
above integral.
The connection of complex multiplication with number theory is nicely described by A. WElL in his essays on number theory [7].
My object in this paper is twofold: I shall give some background and describe
some efforts made to cope with the integral and I shall try, on the basis of earlier
results of FAGNANO, to account for the origins of his transformations--transformations which were to stimulate EULEg.

§ 2. Early Emergence of the Lemniscate
The lemniscate is familiar to all students of calculus. It is the figure eight whose
equation in polar coordinates is given by

1.2 ~ a2 cos 2

and in Cartesian coordinates by

(x 2 + y2)2 :

a2(x 2 __


It was given this name, which is derived from the Greek word for ribbon, by
JAMES BERNOULLI (1654--1705).
Its origin, however, is older. The lemniscate is a special case of the ovals of
CASSINI (GIOVANNI DOMENICO CASSINI, 1625-1712). A CASSlNI oval is the locus
of the intersection of the tangent to a conic and the perpendicular drawn from the
origin to the tangent, if the conic happens to be a rectangular hyperbola, then
the CASSINI oval becomes the lemniscate. The lemniscate is also the locus of a
point which moves so that the product of the distances from two given points is
a constant. From any of these definitions, it is difficult to surmise that the curve
would have any deep significance.
In the years immediately following the publication of NEWTON'S Principia
(I. NEWTON, 1642--1727), and the papers of G. LEIBNITZ (1646--1716) on calculus,
many mathematicians used these new-found techniques to solve diverse mechanical-geometrical problems. These problems arise from trying to find the path of a
particle moving under various constraints. One such problem is the tautochrone
of C. HUYGENS(1629-1695): What is the path of a particle, moving under gravity,
the time of whose descent to a point P is independent of the starting point ?
Professor TRUESDELLhas kindly pointed out that had EULER not been immersed
in studying the integrals arising from the theory of elasticity, he would not have been
so struck by FAGNANO'Sresults.

The Lemniscate, Fagnano, and Elliptic Integrals


The curve in question turns out to be a cycloid. The cycloid was made more
famous by JOHN BERNOULLIwho solved the problem of the brachistochrone, i.e.
the curve with the property that a particle moving under gravity from a point
A to a point B , does so in minimum time.
JOHN BERNOULLI proposed several other problems of which the following,
pertinent to our discussion, is the one referred to by FA6NANO: What is the
curve with the property that the time taken for a particle to traverse the curve is
proportional to the distance from a fixed point. BERNOULLI Called this curve
"the paracentric isochrone". The solution of the problem leads t o the integral


0f -




BERNOULLI was not able to evaluate the integral (which is not now surprising).
He went on to notice, however, that if we consider the curve whose equation is
given parametrically as

x = ~ a t - k t 2,


y = ~ a t - - t 2,

then the integral for the length of arc of this curve is essentially that given in (1)
Indeed, it is not certain whether the problem o f mechanics provided the motivation for this rectification or vice versa, for in the hands of the BERNOULLISand
their contemporaries geometry and mechanics blended into one another, In any
event, eliminating t in (4) gives
(x 2 .+ y2)2 = 2a2(x 2 _ y2)
and this is, again, the lemniscate.
We note parenthetically that if the particle is released from rest at 0 and if
the time to go from 0 to P along the segment OP is equal t o the time along the
arc OP, then the curve C turns out to be a lemniscate.
Finally, as we have remarked above, there is the problem proposed by JAMES
BERNOULLI in 1691: Determine the equation of the curve formed by a flexible
elastic band constrained b y its o w n weight or some attached weight or some
compressing force I J+~ES publishe d his solution in 1694. Taking a special case, he
was led to the integral 1.1.
JAMES BERNOULLIwas not unaware of the connection with the lemniscate. He
found a relation which connects the elastica, the lemniscate and the paracentric
isochrone [8].
Having derived the integral 1.1 governing the rectangular elastica, JAMES
BERNOULLI writes. "I have strong grounds to believe that the construction of
our curve dePends neither on the quadrature nor on the rectification of any conic
Section." He did Show that




-- 1 +


2 n n ! ( 4 n ~ - 1)



§ 3. Elliptic Integrals
The integral (I.1) is one of a class which have come to be called elliptic integrals. Before proceeding we shah recall a few definitions and facts about these
Definition. If R(w, x) is a rational function of w and x, and w2 is a cubic or quartic
function of x, then


f R(w, x) dx

is called an elliptic integral. This can easily be shown to reduce to the case

; T(x) dx,




where T(x) is independent of w. A little more work shows that (2) can be reduced
to one or more of three standard forms named by A. M. LEGENDRE (1752-1833)
integrals of the first, second, and third kinds. The integral


l/1 _ x 4


is an elliptic integral of the first kind, the general form of which is


l/(ax 2 + b) (cx 2 + d)

a, b, c, d being any real numbers not all of which are zero.
Definition. An elliptic function is the inverse function of an elliptic integral.

This is but one of several equivalent definitions of this fundamental idea.
There are cases where there are "natural', limits of integration. In these cases
the definite integrals are called complete elliptic integrals. For example in the
case of the lemniscate integral


x 4

is a complete elliptic integral.
Elliptic integrals arise naturally in connection with the determination of the
length of arc of an ellipse. This gave rise to an elliptic integral of the second
They arose, however, in several other contexts; one interesting example is the
motion of a pendulum.

The Lemniscate, Fagnano, and Elliptic Integrals


Then there is the theory of elasticity. EULER'S work on elasticity began in
1728 and his interest continued throughout his life. EULER wrote several papers on
elasticity before 1751 and these culminated in an outstanding piece of work entitled " O n Elastic Curves" published in 1743 as a supplement to a treatise on the
calculus of variations--"Method of determining curves possessing maximal
or minimal properties". In the paper on elastic curves, EULER determines that
the equation satisfied by the curves assumed by the elastic band has (after a
suitable change of variable) the form


(a 2 -- c 2 -b x 2) d x
]/(c 2 -- x 2) (2a 2 -- c 2 + x 2)

This is the sum of two elliptic integrals, one of the first kind and one of the second.
Now with a masterful blend of analysis, geometry and mechanics, EULER
analyzes the various shapes which the elastic band might assume and finds nine
classes. In the process he notes the existence of a real period for the corresponding
elliptic function, the first such discovery. Of the nine classes, the third is the case
when a = c, which is the rectangular elastica. In this case he expands the complete integral in a rapidly converging power series and refers to a relation between the integrals




o Va*-x

namely A B = ~a2/4. He had communicated this result to JOHN BERNOULU in
1738 with the comment that he had discovered a "singular property of the rectangular elastica". This property of complete elliptic integrals can be viewed as
a special case of LEGENDRE'S identity connecting two kinds of complete elliptic
integrals and is anything but obvious.
In addition to his work on elasticity, EULER published before 1751 several
papers on the rectification of the ellipse. The elliptic integrals arose, of course,
once again. Unable to carry out the integration of these and other integrals,
mathematicians of this era gave numerous transformations and relations among
them. These relations were often given a geometric interpretation.
Thus for example, JAMES BERNOULLI [8] in 1679 showed how to bisect geometrically the arc of a parabolic spiral by reducing the integral to a quadrature.
A year later JOHN BERNOULLIshowed how to construct geometrically a parabolic
arc having a given ratio with a given arc.
FAGNANO was quite familiar with the work of the BERNOULLIS and proved
some interesting relations, of which the following is an important example: If
VI~ @z2
t ~---






+ f - t2- dt !




Geometrically, the integral on the left is the length of arc of a lemniscate; the first
on the right is the length of arc of an ellipse, while the second is that of a rectangular
It is against this background that EULER was to read the works of FAGNANO
and in particular the paper on the lemniscate. It is then understandable that he
would have been quite fascinated by the new and highly original properties of
the lemniscate integral. 2
It should be stressed however, that except for EULER'S 3.5 and his observation
on the periodic character of the integrals, there are no outstanding results on
elliptic integrals prior to FAGNANO'S.
We are naturally tempted to ask whether the claim of JACOBI is justified. It
is generally agreed that the discovery of the addition formula had a great influence on the development of elliptic functions. Equally crucial for this development is the transition from the elliptic integral to its inverse viz. the corresponding
elliptic function. We have remarked above that EULER was close to this transition.
Whether he would have discovered the addition formula without the stimulus of
FAGNANO will remain in the realm of conjecture.
To assert that he would not, would greatly underestimate the creativity of this
great master. On the other hand, to treat the discovery as a routine result would
unfairly detract from the brilliance and ingenuity of FAGNANO. The facts remain
that EULER had not communicated the addition formula before reading the works
of FAGNANO and he did communicate to the Berlin academy on January 27, 1752
(about a month after he got FAGNANO'S work) a paper in which he both extends
and simplifies FAGNANO'S important results.
We turn now to a detailed account of FAGNANO'S contributions to the theory
o f elliptic integrals.

§ 4. A Transformation and Application
Among the transformations discovered by FAGNANO, the following is relevant
to our discussion. We shall consider its genesis in more detail in § 6.

Theorem 4.1. If (u + 1) (v

1) = 2 and



~ '




FAGNANO uses this to bisect the first quadrant arc of a lemniscate. If t is the
square of the distance from the origin to the point P(x, y) on the lemniscate, then
the equation


(x 2 + y2)2
2 See footnote 1.


X 2


The Lemniscate, Fagnano, and Elliptic Integrals


can be parametrized by


t --



~ -'


t 2


The differential o f arc is given by
ds -- m

lit -



Fig. 1



arc O P =-

while by use o f (1) and the substitution (v + 1) (u + 1) ~- 2 we see that



_ l/v-v





I f these two arcs are equal, we get u = v, and hence (u + 1) 2 = 2, U = ¢ 2 - - Solving for x and y, we get

x =

-- 1/2 2


Y =


+ 3 1/22

and these are the required coordinates. It will be observed that, since only square
roots are involved, the points are constructible with straight edge and compasses.
We shall further transform the result o f Theorem 4.1. I f we replace t by t 2
in the parametric equations (4), the differential of arc becomes ds =
and T h e o r e m 4.1 becomes



]/1 -- t 4



Theorem 4.2. If (u 2 ÷ 1)(v 2 + 1) = 2, then
t/1 -


I/1 -


This result will be used in the next section.

§ 5. Fagnano and Complex Multiplication
We come now to the paper entitled "Method for measuring the lemniscate"
which aroused EULER'S enthusiasm. FAGNANO begins by saying that the brothers
BERNOULLI made the lemniscate famous by their studies of the paracentric isochrone. "The determination of the arc of the lemniscate would be important not
only for the isochrone but for other curves as well". The paper has three parts.
In the second, FAaNANO begins with
Theorem F1. "Given the two equations

x =--

1 1/1 ~ i/1








~/1--z 4

I / ~ '

I say that if the first holds, then so does the second."
The proof, he says, is a straightforward verification.
Theorem F 2. "Given the two equations



~/1 - - u 4

1 ¢1- ¢1-z,




1/1 -- z 4



I say that if the first holds, then so does the second.

Proof. Replace x in Theorem F1 by i/1 _ u 4

These are remarkable results, as we shall show. The significance of the second
would have been immediately recognized by EULER.

The Lemniscate, Fagnano, and Elliptic Integrals


This second theorem has two interpretations. The first is geometric. I f O P ---- u
and O Q = z and if u and z are related by (3), then

2 arc O P = arc O Q .

Fig. 2
In other words arc O Q is bisected by the point P. This makes possible the
trisection of the arc O R . FAGNANOaccomplishes this as follows. F r o m Theorem 4.2
if (z 2 + l ) ( u 2 + 1 ) - ~ 2 ,
then arc OP ----arc RQ, while from T h e o r e m F z ,
if u and z are related by (3), arc O P = arc P Q . Eliminating u from these two
equations, we get

z8 +6z ~-3=0.

Solving (6) for the real roots, we get


z = ]/--3 + 2 ]/3.
It is clear that this point can be contructed with straight edge and compasses.
We come to a second interpretation. Let us set






In accord with our previous observation, this is an elliptic integral and what
FAGNANO has proved in Theorem Fz is that if


z =

2u ]/1 - - u '~
1 q-u 4 '



s(z) = 2s(u).

In other words (9) and (10) give a duplication formula for an elliptic integral.
It was not long after reading this paper that EULER proved the following addition
formula. I f

P ( u ) -= 1 + au z - - u 4

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