The Tautochrone of Huygens and Abel:
From Constructive Geometry
to Fractional Calculus

Because astronomical investigations require increasingly accurate time measurements, Huygens devoted considerable effort to the problem. In 1656, he patented the first pendulum clock, significantly improving the precision of timekeeping. This innovation was later described in his 1658 publication Horologium (“The Clock”). Although his investigations into centrifugal force also date back to this period (1659), it was in his major work of 1673, Horologium Oscillatorium sive de motu pendulorum (“The Pendulum Clock, or On the Motion of Pendulums”), that he developed a full theoretical account of pendular motion. It is also in this treatise that the modern formula for the centrifugal force in uniform circular motion first appears. The Horologium Oscillatorium is often regarded as one of the most important work on mechanics prior to Newton’s Principia (1687). In the book, Huygens derived the formula for the period of a simple pendulum under the small-angle approximation — a classical result encountered in the Galileo’s work. Huygens went further by investigating the precise curve along which a pendulum would swing with a period that is truly independent of its amplitude. His solution identified this path as a cycloid.

This remarkable property is known as the tautochronism of the cycloid — from the Greek words ταυτό (equal, same) and ϰρόνος (time) — and it requires that the pendulum follow an inverted cycloidal trajectory. The tautochrone problem is addressed in Part II, specifically in Propositions XVI through XXVI. Let us now proceed by sketching some of the representative steps of his derivation, focusing the last two propositions, starting with Proposition XXV. It states that [4]:

Proposition XXV

In a cycloid with a vertical axis, and with the vertex seen to be the lowest point, the times of descent for some body, on leaving any point on the cycloid from rest until it reaches the lowest point at the vertex, are equal to each other; and this time has the same ratio to the time of fall along the whole axis of the cycloid as the semi-circumference of the circle to the diameter.

To reproduce his construction, we need the help of Fig. 1, part of the Huygens’ book [5].

For a better understanding of the discussion that follows, here is a summary of the elements that appear in Fig. 1 and their meaning:

• •

  Point A — is the lower vertex of the cycloid, the endpoint of the motion;

• •

  BA is an arc of the cycloid — the path followed by a body in free fall;

• •

  Vertical axis DA — a straight line along which the ‘standard’ time is measured for comparison of the motion;

• •

  Semicircle FHA — constructed on the diameter FA, used to establish the proportion of times;

• •

  Tangent BG — the line tangent to the cycloid at point B, parallel to the line EA, which intersects the semicircle.

The cycloid shall be ABC the vertex of which is seen to be A pointing downwards, with the vertical axis truly AD, and by taking some point on the cycloid, such as B, a body is free to fall from natural forces along the arc BA, or along a surface thus curved into this shape. Huygens says that the time of descent of this body to be as to the time of fall along the axis DA, thus as the semi-circumference of the circle to the diameter. From which demonstration, the time of descent along any arc of the cycloid to finish at A also, are agreed to be equal to each other [6]. Huygens then states that the descent time of this body is, in relation to the fall time along the DA axis, as the semicircle is in relation to the diameter. This proportion stated by Huygens — between the descent time on the cycloid and the vertical fall time along the axis — is the expression of tautochronism: in fact, the proportionality constant is $\pi/2$, since the perimeter of the semicircle is $\pi\,r$, while its diameter is $2r$.

The discussion continues by diving into the details of the geometric construction. Huygens explains that a semicircle is drawn on the axis DA. This semicircle intersects the line BF at point E, which is parallel to the base DC. The line EA is then drawn, and the line BG is drawn parallel to it, which turns out to be a tangent to the cycloid at point B. In fact, this line also intersects the line drawn horizontally from point A at point G. Additionally, a semicircle FHA is drawn on the line FA. With this setup, Huygens states that the time it takes to descend along the arc of the cycloid BA is to the time of uniform motion along the line BG (with half the speed of BG) as the arc of the semicircle FHA is to the straight line FA. In other words, the time for uniform motion along BG is the same as the time it would take to descend along the same line BG, or along EA (which is parallel and equal to BG), or equivalently, the time to fall under gravity along the axis DA, as shown in Proposition 6 by Galileo [7]. Therefore, the time along the arc BA is related to the time of descent along the axis DA in the same way the circumference of the semicircle FHA is related to the diameter FA.

The last proposition dedicated to this problem also refers to Fig. 1 and is stated as follows [8]:

Proposition XXVI With the same positions, if some line HI is drawn above which cuts the arc BA in I, and the circumference FHA in H: I say that the time to pass through the arc BI, to the time to cross the arc IA after BI, has the ratio which the arc of the circumference FH has to HA.

The Huygens’ explanation of the construction may be paraphrased as follows [9]. Consider the straight line HI intersecting the tangent BG at point K, and the axis DA at point L. The time it takes to travel along the arc BA is compared to the time it takes for uniform motion along BG with half the speed of BG. This comparison is similar to how the arc FHA compares to the straight line FA. The time of uniform motion along BG is compared to the time of uniform motion along BK, with the same half-speed of BG. The ratio of these times is the same as the ratio of BG to BK, which is equivalent to the ratio of FA to FL. The time for uniform motion with the same speed along BK is compared to the time it takes to move along the arc BI, which is similar to how the length FL compares to the arc FH.

As a result, the time to travel along the arc BA is equal to the time to travel along the arc BI, and this is in the same proportion as the arc FHA is to the arc FH. Finally, by dividing and converting, the ratio of the time along BI to the time along IA after BI is the same as the ratio of the arc FH to HA. We underline that both propositions end by the Q.E.D. statement in the original work because for Huygens the geometrical constructions he employs are rigorous method to demonstrate the validity of a physical result. This is indeed the case as we can show hereafter by recalling the problem using simpler language and more updated mathematical notation.

To proceed this way, let’s first remember that Huygens started with the geometric generation of the cycloid, i.e., he considered a circle of radius $r$ rolling on a horizontal surface without slipping. A fixed point on the edge of the circle traces a path that defines the cycloid. The resulting curve is symmetric, with its lowest point at the vertex. When the curve is inverted (with the concavity facing upwards), it gives the ideal profile of a tautochrone pendulum. Huygens’ analysis leads us to think that, for the descent time along a curve to be independent of the starting point, the curve must have the property of equating the acceleration of a body along it with the appropriate free fall acceleration for comparison purposes.

Using a modern approach, we could say that he wants to find a curve such that

where $ds$ is an element of the curve, $g$ is the acceleration due to gravity, and $y$ denotes the vertical axis in a Cartesian system. In this approach, the deduction is based on comparing two motions, namely, the motion along the inverted cycloid (frictionless sliding) and uniform circular motion, i.e., motion with constant speed along a circle. Based on these premises, Huygens establishes a correspondence between the vertical height of the particle on the cycloid and the angular position of a point moving uniformly along a circle. He then shows that, if one considers an auxiliary circle that generates the cycloid (by rolling), the time it takes for a point to reach a certain height on the cycloid coincides with the time a point in uniform motion takes to reach the same angle on the circle — provided the variables are correctly associated. In a few words, the projection of uniform circular motion along the generatrix of the cycloid coincides with the free fall motion along the cycloid — and since uniform circular motion is isochronous, the motion along the cycloid will also be isochronous!

In proceeding, Huygens notes that a common pendulum (one suspended by an ideal string) is not isochronous: its period depends on the amplitude of the oscillation. To resolve this, he proposes restricting the motion of the string so that the pendulum, when oscillating, follows an inverted cycloid rather than a circular arc. This ensures that, regardless of the amplitude, the oscillation period of the pendulum remains exactly constant. With these ingredients, Huygens then demonstrates that the inverted cycloid is the only curve for which the descent time is independent of the starting point. In fact, the descent time from any point to the vertex is

which coincides with the period of a cycloidal pendulum. Finally, the ratio between this time and the free fall time along the cycloid’s axis is given by

In other words and somewhat more formal terms, we wish to find the curve $y(x)$ such that

where $v(y)$ is the velocity of the particle along $y$. In this case, we have

and, also, the velocity is given by $v(y)=\sqrt{2gy}$, a result derived using the conservation of the mechanical energy. Thus, the descent time for the particle can be defined as

which is the same integral that appears in the brachistochrone problem — the curve of minimum time between two points.

The starting point of this investigation, in fact, was a challenge posed by Johann Bernoulli in 1696. It consisted of finding the trajectory followed by a body starting from rest at a given point, in a gravitational field, along a frictionless curve, until the final point. The curve is known as the brachistochrone, from Greek: βράϰιςτος (shortest) and ϰρόνος (time). The problem was tackled (and solved) by Jakob I Bernoulli, in an article dedicated to solving one of the problems proposed by his brother: Solutio problematum fraternorum (Solution to the brother’s problem), which he titled De curva celerrimi descensus (literally: the curve of fastest descent) [10]. Euler later solved the same problem, thereby founding the calculus of variations [11, 12]. Later, Lagrange revolutionized the field by introducing the $\delta$ algorithm and incorporated variational methods into a much more general mathematical framework [13, 14].

As can be seen from the excerpts of Huygens’ demonstration, none of these equations were employed in establishing his results. We have introduced them here merely to bring Huygens’ language closer to the one we use in the study of the fundamental problems of mechanics. This fact, by itself, illustrates the importance that analytical methods have played in mechanics. Indeed, in the period between the publication of Newton’s Principia in 1687 and the emergence of Lagrange’s Méchanique Analytique in 1788, a highly elaborate language in mathematical physics developed — an outcome of the effort to integrate mechanics into mathematical analysis. It was a fertile period during which scientists of great stature, such as Leibniz, Varignon, Jacob I and Johann I Bernoulli (and his son Daniel), Euler, d’Alembert, Laplace and Lagrange, among others, devoted themselves wholeheartedly to this endeavor. The work of these men led to the emergence of analytical mechanics, practically in the form we still know and that is presented in our textbooks today.

On the first page of Abel’s 1823 article, dedicated to definite integrals, we read [17]:

It is well known that, with the help of definite integrals, many problems can be solved which otherwise cannot be resolved, or at least are very difficult to handle […] I will demonstrate a new application by solving the following problem.

Abel then considers a problem more general than the classical tautochrone, which we now briefly summarise in order to follow the core ideas and consequences of his approach. Consider Fig. 2, which illustrates the motion of a particle along an arc, such as the one described by the curve CA. Let CB be a horizontal line and AB perpendicular to CB. The curve AM is described in Cartesian coordinates, where $AP=x$ and $PM=y$. We also define $AB=a$ and denote by $AM=s$ the arc length along the curve, which is initially unknown.

The particle begins its motion from rest at point C and moves along the curve CA. The time $T$ required to traverse the arc depends both on the shape of the curve and on the distance $a$. The problem is thus to determine the form of the curve KCA such that the time of descent $T$ is a given function of $a$, for instance, $T=\psi(a)$. We use today’s notation; the notation employed by Abel is such that a function $\psi(a)$ is denoted by $\psi\,a$.

After some mechanical considerations, Abel found for this problem the following equation defining the time involved in the motion of the particle:

in which $s=s(x)$ is the unknown function. However, instead to handle with it directly, Abel considers the more general equation:

where $n$ has to be less than 1 to prevent the infinite integral between two limits; $\psi(a)$ is an arbitrary function that is not infinite, when $a=0$.

To proceed, let us recall that the problem is to find the curve $s(x)$ which Abel searches in the form of a power series

After term-by-term operations with the series, and after using the properties of the gamma function — by quoting in this regard the work of A. M. Legendre —, and after some pages of manipulations with the power series, he arrives at the following solution to Eq. (9), called by him as a “remarkable theorem” [20]:

In a few words, the remarkable result is that given an integral like the one in Eq. (9), the curve $s(x)$ is given by Eq. (11). After examining the obtained formulas from other perspectives, Abel mentions that $s(x)$ can be expressed in a different way, namely:

As underlined in Ref. [20], it is easy to recognize, behind the archaic notation he used, that Abel presents two expressions for the fractional-order integral: a derivative of negative order and a symbol that later appears in Liouville’s works for the integral of order $n$: $\int^{n}\psi(x)dx^{n}$ (See Refs. [21, 22, 23]).

Subsequently, Abel considers the case $n=1/2$, in order to obtain from the integral

the desired curve $s(x)$, when the time is given by $psi(a)$, i.e.,

from which one obtains:

For our analysis, this is a crucial point in the paper because Abel inverts the fractional integral by using a fractional differentiation on both sides of the equation, thus assuming (and understanding) that fractional order integration and differentiation are mutually inverse procedures. He affirms indeed [17]:

If the equation of a curve is $s=\psi(x)$, the time a body takes to travel along an arc of the curve whose height is $a$ is $\sqrt{\pi}\frac{d^{1/2}\psi(a)}{da^{1/2}}$.

To conclude this incursion for the original approach of Abel, we remark for future purposes that Eq. (9) could be rewritten as:

which, apart from a constant coefficient, may be identified with the well-known Caputo’s time-fractional derivative of real order to be introduced later [24, 25]. In this perspective, the solution proposed by Abel is the inverse operation of the fractional integral of the same arbitrary order $n$ [20].

To appreciate the result obtained by Abel in more simple terms, let us first reformulate the physical problem of a particle of mass $m$ sliding the curve shown in Fig. 3. It is released from the rest at the position $y=y_{0}$ and slides under the influence of the gravity, along a frictionless wire.

The problem is to determine the curve $s(y)$, followed by the particle, such that the time employed to slide down to the lowest point under gravity is independent of its initial position ($x_{0},y_{0}$) on the curve. If the mass is released from $y=y_{0}>0$, then its speed at $y$ will be given by

from which we obtain:

because $ds/dy>0$, but $dy/dt<0$. The time of descent is then

We observe that (19) was obtained by invoking only the conservation of the mechanical energy, Eq. (17). It is an integral equation to be solved in order to get $f(y)=ds/dy$. Let us change a little the notation ($y_{0}\to y,\,y\to z$) to search for a curve $s(z)$ such that

The formal problem is then to find $f(z)=ds/dz$, given $k(y)$:

which may be recognized as a convolution between $f(z)$ and a kernel $K(y,z)=(y-z)^{-1/2}$. Equation (21) is an integral equation for $f(z)$ and may be solved by taking the Laplace transform of its both sides. Remembering that the Laplace transform is defined as

where $\mathbb{C}$ stands for the set of complex numbers, after its computation, we obtain

where $c_{1}=k(y)/\sqrt{\pi}$ is a constant. The inverse transform of $F(s)$ is

where $c$ is another constant. Remember that $f(z)=ds/dz$ or $f(y)=ds/dy$. Thus,

By putting $y=c^{2}\sin^{2}(1/2\theta)$, a short calculation yields

which are the parametric equations of a cycloid [26].

Let us pause for a moment to underline a central point of the present analysis. We remember that the integral equation obtained by Abel in the tautochrone problem can formally be written as

However, this is, essentially, the definition of a fractional integral of order $1/2$, in the sense of Riemann–Liouvile, as we discuss soon. We may rewrite it in the form:

where we have introduced the notation ${}_{0}{\textrm{I}}_{x}^{1/2}$ for an integration of arbitrary order $1/2$ in the variable $x$. Thus, the tautochrone problem in the Abel’s perspective may assume the form:

In this way as well, to obtain the unknown function $g(x)$ became possible if we promote the inversion of a fractional operator, i.e., of the fractional integral of order $1/2$. In this framework, the Abel problem would be the particular case of the inversion of a fractional operator. The formal solution would require the application of the corresponding fractional derivative, i.e.,

This kind of simplified reasoning opens space to discuss the fractional calculus as a natural generalization of the classical differential operators, motivated by historical problems. Thus, the modern interpretation of the Abel problem as involving a fractional operator is an instructive bridge between the early years of the mechanics and mathematical analysis and the present.

Let us now move to a more formal way to connect again these results from Abel to the roots of the fractional calculus. To accomplish this task, the problem can be reformulated in such a way that one has to find $f(z)=ds/dz$, given $k(y)=k$, such that:

Now, we proceed in a different manner, multiplying both sides of (31) by $1/\Gamma(1/2)=1/\sqrt{\pi}$ in order to obtain

whose right-hand side we can recognize (better: we define!) as a semi-integral operator, that is, a fractional integral operator of order $1/2$, represented by the symbol ${}_{0}{\textrm{I}}_{z}^{1/2}$ we have introduced before. If this is true, to cancel the “semi-operator”, we have just to take the “semi-derivative” of each side of (32), that is,

In this way, the problem is reduced to calculate the derivative of order $1/2$ of the left-hand side, i.e., of a constant. This calculation may be performed by using a result due to Lacroix [27].

Indeed, a derivative of arbitrary order was briefly mentioned (perhaps for the first time as such) in 1819, in the book of Sylvestre François Lacroix (1765–1843), from which an explicit formula for a fractional derivative may be obtained. The central argument may be summarized as follows (using a today’s notation). Let, for instance, $y=x^{m}$; when $n$ is an integer, one has for any arbitrary $m\in\mathbb{N}$:

where $\mathbb{N}=\{1,2,3,...\}$ is the set of natural numbers. If we put $m=2$ and $n=1$, then we obtain

as expected. Now, we may consider $m=1$ and $n=1/2$ and try to answer to the original question. We obtain:

Thus, the result obtained by Lacroix may be put in the usual form as:

which, apart from a constant, may be compared with the result reported by Leibniz, in his letter dated September 30, 1695, to L’Hôpital [28]:

You see from this, Sir, that one can express by an infinite series a quantity such as $d^{1/2}\overline{xy}$, or $d^{1:2}\overline{xy}$, even though this may seem far removed from Geometry, which ordinarily knows only differences with positive integer exponents, or negative ones with respect to sums, and not yet those whose exponents are fractional. It is true that it remains to provide $d^{1:2}x$ for that series; but even this can be explained in some fashion. For let the ordinates $x$ be in geometric progression, so that, taking a constant $d\beta$, one has $dx=xd\beta:a$, or (taking $a$ as unity) $dx=xd\beta$. Then $ddx$ will be $x\overline{d\beta}^{2}$, and $dx^{3}$ will be $=x\overline{d\beta}^{3}$, etc., and $d^{e}x=x\cdot\overline{d\beta}^{e}$. And by this device, the potential exponent, and replacing $dx:x$ with $d\beta$, we have $d^{e}x=\overline{dx:x}^{e}\cdot x$. Thus it follows that $d^{1:2}x$ will be equal to $x\cdot\sqrt[2]{dx:x}$. It seems likely that one day useful consequences will be drawn from these seemingly useless paradoxes. You are among those who can go furthest in discoveries, and I shall soon be obliged to hand over the torch to others. I wish I had much to share, for this verse: your knowledge is nothing unless someone else knows that you know is most true, in that thoughts which were of little value in themselves can give rise to far more beautiful ones.

Thus, the result presented by Leibniz for the question raised by L’Hôpital may be interpreted as stating that for the fractional exponent $e$, one has:

which could be compared with Eq. (34), because, for $m=1$ and $n=e$, it could be rewritten as:

In this framework, if we consider consider the case $n=e=1/2$ and $m=0$, that is, $y=x^{0}=1$, from Eq. (34), we obtain the “surprising” result that the fractional derivative of a constant is not zero, but instead a function of $x$, namely:

Coming back to our problem, by using the results above in Eq. (33), we obtain:

This permits us to deduce that the answer to the tautochrone problem is formally:

Integrating the above equation, using the condition $s(0)=0$, we obtain:

which is the cycloid sketched in Fig. 3, as shown by Abel in connection with this problem of finding the path followed by the particle in order the time of sliding down be independent of the initial height [17].

In summary, to solve the tautochrone problem, in the pioneering Abel’s perspective, we have to determine $f(z)$, a task that can be accomplished by computing the fractional derivative of a constant $k$. In the paper of Abel, it is not clear that if he used the Lacroix or its own results. Anyway, in the paper of 1826, Abel provided the solution for the integral equation [18]:

In practice, Abel used the operators that nowadays are ascribed to Riemann and Liouville, preceding them by at least one decade [29, 30]. To complete the analysis, let us formalize a little bit more the previous results, considering the Abel fractional integral equation of first kind, defined as:

Equation (21) is a particular case of Eq. (45), for $\alpha=1/2$. As we have seen above, Eq. (45) may be solved in terms of a fractional derivative in the form:

where we have also introduced the symbol ${}_{0}{\textrm{D}}_{t}^{\alpha}$ now to denote a derivative of arbitrary order $\alpha$ in the variable $t$. A notation of this kind was introduced by the mathematician Harold Thayer Davis (1892–1974) [31] and is useful to highlight the meaning of the left subscript, which is not obvious in a derivative [32].

In Eq. (46), we have supposed that ${}_{0}{\textrm{D}}_{t}^{\alpha}$ exists (even if we have not discovered its form until now!) and is the left-inverse of the operator ${}_{0}{\textrm{I}}_{t}^{\alpha}$, that is,

where $\mathds{1}$ is the identity operator. A solution for (45) may be now searched by using the Laplace transform and the convolution theorem [32, 33].

L. R. E. gratefully acknowledges the hospitality of the Department of Molecular Sciences and Nanosystems at Ca’ Foscari University of Venice, Italy, and the Department of Materials, Environmental Sciences and Urban Planning (SIMAU) at the Polytechnic University of Marche (UnivPM), Ancona, Italy, where he served as a Visiting Scholar during the preparation of this work.
The research of F. M. has been carried out in the framework of the activities of the Italian National Group of Mathematical Physics (GNFM-INdAM).

No specific financial support was required for the work. L. R. E. acknowledges the continuous partial financial support from the Brazilian Agency — CNPq, by means of the Grant n. 30295/2022-9.

Both authors contributed equally to the final version of this paper. The initial idea was suggested by F.M. The conceptual framework was developed after a fruitful meeting in Pisa in 2024. The first draft was written by L.R.E. Refinements and the first complete version emerged after an extended and productive exchange of correspondence between the two authors. The current and final form of the paper is the result of a joint effort, consolidated during a personal meeting in Rimini — land of Francesca and Paola Malatesta, whose tragic love was immortalized by Dante in the Divine Comedy — on the shores of the Adriatic Sea.

The authors declare no potential competing interests.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

This manuscript benefited from the use of Generative AI tools (ChatGPT) for translation assistance and grammar refinement, given that the authors are not native speakers of English.

Luis Roberto Evangelista is Professor of Theoretical Physics at the PhD Preogram in Physics at the University of Maringá (UEM), Brazil. His research interest lie in complex fluids, complex systems, and history of physics, and include mathematical physics of liquid-crystals, diffusion problems, and adsorption-desorption phenomena.
His Orcid Number is 0000-0001-7202-1386

Francesco Mainardi is Professor (Retired) of Mathematical Physics at the University of Bologna, Italy, renowned for his pioneering work in fractional calculus. His research focuses on fractional differential equations, viscoelasticity, and anomalous diffusion. He played a key role in promoting the Mittag-Leffler and Wright functions in applied contexts. Mainardi authored influential books, including Fractional Calculus and Waves in Linear Viscoelasticity. He remains an active voice in the field, shaping theory and applications across physics and engineering.
His Orcid Number is 0000-0003-4858-7309