Lambert, the Circle-Squarers and $\pi$: Introduction to Lambert’s Vorläufige Kenntnisse

Abstract

The treatise «Preliminary Knowledge for Those Seeking the Quadrature and Rectification of the Circle» («Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen») was written in 1766 and published in 1770 as part of the second of the three volumes entitled Contributions to the Use of Mathematics and Its Application (Beyträge zum Gebrauche der Mathematik und deren Anwendung).

Twenty-two years I have been trying to find the fixed point. [...] The same thing happens to me with the quadrature of the circle, which I have been so close to finding, that I do not know, nor can I conceive, how I do not already have it in my pocket.

—Miguel de Cervantes, The Dialogue of the Dogs.

[T]he pursuit of mathematics is a divine madness of the human spirit.

—Alfred North Whitehead, Science and the Modern World.

The  treatise «Preliminary Knowledge for Those Seeking the Quadrature and Rectification of the Circle» («Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen») was written in 1766 and published in 1770 as part of the second of the three volumes entitled Contributions to the Use of Mathematics and Its Application (Beyträge zum Gebrauche der Mathematik und deren Anwendung).¹ According to what Lambert wrote in the preface to this volume, his treatise was intended for «searchers for the quadrature of the circle»,² i.e., those who sought to square the circle «with straight edge and compasses» or, ultimately, by means of algebraic curves.³

Now, unlike the memoir that Lambert presented the following year and in which he included the —as far as we know— first rigorous demonstration of the irrationality of $\pi$, his treatise of 1766 is a «semipopular and witty exposition»⁴ about the impossibility of obtaining a finite decimal representation of $\pi$. Eventually, with the proof of the transcendence of $\pi$, the impossibility of constructing —using only compass and ruler— a square the area $x^{2}$ of which is equal to the area $r^{2}\pi$ of a circle, with $x=r\sqrt{\pi }$, was to be established.⁵ But a century before this, in the present work, Lambert shows the impossibility of expressing the ratio of the diameter to the circumference «by means of a rational fraction», which was a key aim of those who were known at the time as «circle-squarers». Therefore, while at the time the adjective «preliminary» was used to describe a text that provided a very first approach on a topic, in this case the expression «preliminary knowledge» (for the circle-squarers) may already have been indicative of the ironic tone of the text.

In the late 17th and early 18th centuries there was a burgeoning of attempts to solve the classical problem of the commensurability of the ratio of the circumference to the diameter of the circle. This was largely due to the spreading of the notion that certain states and scientific institutions were awarding prizes for the resolution of this geometrical problem because of its presumed relevance to the resolution of the problem of determining the «fixed point», or the longitude at sea. As Augustus De Morgan pointed out, such was the number of attempts at the time that from the mid-18th century onwards some institutions, including the French Académie Royale des Sciences and the Royal Society of London, decided not to examine any further work on the subject.⁶

At the time, the importance of determining longitude at sea resided, first and foremost, in its economic and political consequences. In the face of overseas expansion and the ensuing development of maritime trade, some states, scientific institutions, companies and individuals did in fact start to offer prizes for the resolution of this problem from the second half of the 16th century on.⁷ This led to the development of new methods, techniques and instruments, and eventually contributed to the more accurate calculation of longitude at sea from lunar distances (with better observational and navigational instruments, as well as improved tables of the distance between the moon and other celestial bodies) and the construction of more precise marine timepieces.⁸ But this also led many people to associate the problem of longitude at sea with that of the quadrature of the circle, since —at best— it was assumed that solving the latter would contribute to solving the former, for example by improving the instruments known as «quadrants», such as the quadrant of reduction or sinical quadrant (see Fig. 2.1), which were commonly used to tackle the problem of longitude at sea.⁹

Fig. 2.1

Quartier de réduction (1671), engraving by François Jollain (available at https://data.bnf.fr/atelier/14952804/francois_jollain/)

The confusion which ensued here was at least partly due to the fact that some encyclopaedias and dictionaries spread the idea that some of those states which were offering prizes for solutions to the problem of longitude at sea were in fact offering these prizes just for solving the problem of the quadrature of the circle.¹⁰ This was not the case, even though it is true that some prizes linked to this latter problem were indeed offered. Thus, while there were people who offered prizes for the refutation of the alleged solutions to this problem,¹¹ Jean-Baptiste Rouillé de Meslay allocated a sum of money for research about it in his will, although the French Académie Royale des Sciences ended up re-allocating it to research on navigation.¹²

The works mentioned by Lambert in §. 3 precisely form part of the corpus produced by circle-squarers during the 18th century. Moreover, these works not only coincide in their purpose but also in the specific ratio which they proposed for the circumference of a circle to its diameter, namely 3844 : 1225, obtained from $\frac{4\cdot 35^{2}}{31^{2}}$. It follows, then, as Lambert notes (§§. 2–3), that Joseph Ignatius Carl von Leistner (in 1737 and 1740), Johann Christoph Merkel (in 1751) and, following the latter, Johann Christoph Bischof (in 1765) all advocated a much less precise approximation of $\pi$ than others well known at the time, starting with that by Archimedes,¹³ according to whom

$$ 3+\displaystyle \frac{10}{71}<\pi <3+\displaystyle \frac{1}{7}, $$

which dates from the third century BCE¹⁴ and is accurate to two decimal places, or that by Ludolph van Ceulen ¹⁵ for the first 32 decimal places of the lower and upper bounds of $\pi$:

$$\begin{aligned} \text {Lower bound}\; &:&\; 3\displaystyle \frac{14159265358979323846264338327950}{100000000000000000000000000000000}\\\\ \text {Upper bound}\; &:&\; 3\displaystyle \frac{14159265358979323846264338327951}{100000000000000000000000000000000}. \end{aligned}$$

In order to discourage the use of such a popular ratio, Lambert explains a «general rule» which might be used to obtain more accurate ratios. First, he says, given $a=1$ as the diameter of the circle and b as the side of a square with the same area as the former, i.e., $\pi \cdot r^{2}=\frac{\pi \cdot a^{2}}{4}=b^{2}$, one gets $a^{2}:4b^{2}=1:\pi$ and therefore $a:b=2:\sqrt{\pi }$ or $= \frac{200000000000}{1.77245385075}$. Secondly, he calculates the continued fraction associated with the decimal development of this quotient up to the seventh row and from this he obtains the sequence of rationals $\frac{b}{a}=\frac{7}{8}, \frac{8}{9}, \frac{31}{35}, \frac{39}{44}, \frac{109}{123}, \frac{148}{167}, \frac{3845}{4342}$, etc., which are «more precise according to their order» (§. 4).

As Lambert points out, such fractions express the side of a square with the same area as a circle, the diameter of which is assumed to be $= 1$, so that, inverted, they express the diameter of a circle the area of which is $= 1$ (§. 5), thus providing a calculation the margin of error of which is insignificant for certain practical matters: for such a circle, the diameter of which is 1.128379..., the fraction $\frac{35}{31}$ approximates with a difference of 0.00065..., the fraction $\frac{44}{39}$ approximates with a difference of 0.00017..., and so on. Finally, he addresses the case of cube numbers, used for the comparison of the diameter of a sphere with the side of a cube with the same volume as the former, obtaining the sequence of rationals from the continued fraction associated to $\root 3 \of {\frac{\pi }{6}}$ (§. 6), and explains the procedure which he followed in order to obtain this cube root and to verify that it is correct up to the 18th decimal place (§§. 8–9).

Lambert then goes on to discuss the problem of «whether the ratio of the diameter to the circumference can be expressed by means of a rational fraction» (§. 10) and presents a sequence of 27 ratios obtained following a method used in his treatise «Transformation of Fractions» («Verwandlung der Brüche», also included in the second volume of his Contributions).¹⁶ As he explains, each of these ratios «is more exact than the preceding one» (§. 10), which means that any rational proposed as the exact value for the ratio of the diameter of a circle to its circumference should therefore be greater than the last ratio provided by him, namely $\frac{1019514486099146}{324521540032945}$, which, without this affecting his argument, is incorrect along with his 26th ratio. Lambert himself acknowledges that his ratios correspond to the Ludolphian «numbers» only up to the 25th decimal place and gives the continued fraction from which he obtained his 27 ratios, noting that in his other treatise he gives a continued fraction «which continues to infinity, according to a certain law, and completely removes the hope of determining the ratio of the diameter to the circumference by means of whole numbers» (§. 10).

Precisely, after showing that neither e nor $e^{x}$, with x rational, «can be expressed exactly by a rational» (§. 11), Lambert presents in §. 12 the continued fraction for the function $\tan v$:

$$ \tan v=\frac{1}{1:v-\frac{1}{3:v-\frac{1}{5:v-\frac{1}{7:v-\frac{1}{9:v-\,\,\text {etc.}}}}}} $$

from which, given an integer n and $v=\frac{1}{n}$, he obtains:

$$ \tan v = \frac{1}{n-\frac{1}{3n-\frac{1}{5n-\frac{1}{7n-\frac{1}{9n-\frac{1}{11n-\,\,\text {etc.}}}}}}} $$

As Lambert notes, since this fraction continues to infinity, the tangent of a rational circular arc «will necessarily be irrational» (§. 12), a conclusion which he explains in detail in the subsequent sections (§§. 12–14). In addition, he notes that from this it follows that, «conversely, the arc of every rational tangent is irrational» (§. 15), as well as that, in the case of the arc of $45^{\circ }$, it has «no rational ratio to the radius of the circle» (§. 16). Lambert did not explain further the consequences of his results, but what was entailed by these latter was that $\frac{\pi }{4}$ was irrational and, ultimately, that $\pi$ was irrational as well. In fact, in his treatise «Transformation of Fractions» he had already given the continued fraction for the arc of $45^{\circ }$, which he obtained from the series $\arctan z = z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\frac{z^{9}}{9}-\text {etc.}$¹⁷

Finally, Lambert goes on to state the impossibility of the radius, the arc and the tangent being all commensurable at the same time, so that if the latter «have a rational ratio to each other, then both are incommensurable with the radius» (§. 17), and presents to his readers the following «phenomenon» in §. 18:

if one divides 1 by 0, 7853981634..., as a fourth part of the Ludolphian numbers, it occurs 1 time and subtracts 0, 2146018366... If one further divides by this remainder 0, 7853981634..., which was previously the divisor, then it occurs 3 times and subtracts 0, 1415926536... If one places the number 3 in front of this remainder, one gets 3, 1415926536..., which are precisely the Ludolphian numbers.

Lambert does not explain the «cause» of this «phenomenon» and merely warns that nothing can be concluded from it regarding the quadrature of the circle, the point here being rather just that, given $\pi =3+x$, in order to find x one sets $\frac{1}{\frac{\pi }{4}}=1+\frac{r}{\frac{\pi }{4}}$, from which $r=1-\frac{\pi }{4}$, and $\frac{\frac{\pi }{4}}{r}=3+\frac{x}{r}$, from which $x=\frac{\pi }{4}-3r$, so that $x=\frac{\pi }{4}-3+\frac{3\pi }{4}$ and, therefore, $x=\pi -3$.¹⁸

Over the next hundred years, frequent attempts continued to be made to square the circle and to find the rational value of $\pi$. These included the following notorious cases: during the 1770s Alexandre-Henry-Guillaume le Roberger de Vausenville made several attempts to have his quadrature of the circle either recognised or refuted and even went so far as to sue the French Académie Royale des Sciences and demand that the aforementioned prize instituted in honour of de Meslay be awarded to him;¹⁹ in 1836 Joseph LaComme, a peasant artisan who, in trying to determine «the amount of stone required to pave the circular bottom of a well», found out about the problem of the ratio of the diameter of a circle and its circumference, decided to learn mathematics on his own and to focus fully on solving this problem, coming to the conclusion that the exact ratio was 25 : 8, a result for which he eventually achieved some recognition;²⁰ and from 1859 on James Smith not only published several works in which he also asserted that the «true value» of such a ratio was 3.125, but in addition to this corresponded with several mathematicians, such as De Morgan and William Whewell, who tried, unsuccessfully, to make him see his error, as he himself revealed by publishing some of these letters.²¹

Moreover, even after Carl Louis Ferdinand von Lindemann’s 1882 proof of the transcendence of $\pi$ (i.e., that it cannot be the root of a polynomial with rational coefficients and therefore it is not possible to square the circle «with straight edge and compasses»), there continued to be claims raised about the rationality of $\pi$, such as $3+\frac{13}{81}$ in 1934 and 3.1428 in 1983,²² and about the possibility of the quadrature of the circle. However, whereas in the first case it has been established that this is a perennial quest, one equated in Cervantes’ The Dialogue of the Dogs with the punishments of Tantalus and Sisyphus, in the second case a number of results obtained during the last three decades illustrate how the reformulation of a problem within a different framework can lead to new ideas: Tarski’s circle-squaring problem, for example, asks if a disc in the plane $\mathbb {R}^{2}$ is equidecomposable with a square of the same area (i.e., if the former can be decomposed into finitely many pieces which can be reassembled to obtain a partition of the latter), and Laczkovich (1990), Grabowski et al. (2016/2020), Marks et al. (2017) and Máthé et al. (2022) have all proven in different ways that it is possible to do so. For better or for worse, then, a certain venturing into endeavours deemed impossible is not only inherent to mathematical practice but actually enriches this latter. Lambert’s own work is an example of this.