John Wallis and the “Rectification” of the Parabola

Abstract

The problem of the rectification of curves and the calculation (for which we shall adopt the more geometric term “flattening”) of the surface of a rotating solid generated by a curve, once the rotation axis has been fixed, were treated by John Wallis in two short inter-related tracts on cycloids and cissoids in 1659. In this work, we focus on the problem of the rectification of the parabola, an ancient problem reproposed in the 17th century, and on the solution obtained by Wallis using particular series that reflect the mechanical method of Archimedes, even if this is not immediately apparent to the reader. In presenting Wallis’s speech, we use the language closest to us and make explicit all the calculations not performed by the author, favoring an understanding of the part of his writing dedicated to the rectification of the parabola.

1. Introduction

Once familiar with the classical methods of solving problems, 17th century mathematicians identified new ones, which they approached with new methods. The most relevant ones concern the calculation of the tangent (a direct and inverse problem) on a curve at a point, that of the quadrature (calculation of the areas delimited by curves and that of the volumes of the solids of rotation), that of the length of a curve (indicated by the term ‘rectification’), and that of the surface of the solids of rotation (indicated hereinafter with the term “flattening”).

On the problems regarding the calculation of the tangent and that of the quadrature, numerous studies were performed throughout the course of the 17th century. Some works in which these issues are addressed are [1,2,3,4,5,6,7]. The problems of rectification and flattening, on the other hand, find more space after the development of the calculus applied to geometry. A presentation of the historical evolution of the construction concept and the construction–rectification relationship of the curves can be found in [8].

In this work, we focus on the way in which John Wallis, Savilian professor of geometry at Oxford University from 1649 to 1703, tackles the problems of rectification and flattening by analyzing some pages of his two short tracts (1659) on the cycloid and cissoid, which we shall consider as a single treatment given the uniqueness of the problem.

The title of Wallis’ treatise is: Tractatus duo. Prior, de Cicloide et corporibus inde genitis. Posterior, Epistolaris; in qua agitur, de Cissoide, et Corporibus inde genitis, et de Curvarum, tum Linearum Ευθυνσει (Euthunsei), tum Superficierum Πλατυσμϖ (Platusmo). Anno 1659 editi (Two treatises. The first, on the cycloid and on the bodies it generates. The second, epistolary, which discusses the cyssoid, and the bodies generated by it, and the curves, both the straightening of the lines and the flattening of surfaces. Published in 1659).

The two treatises can be found in [9] (pp. 489–569). The original edition with the same title was printed in Oxford in 1659 as an independent volume.

This work first deals with the cycloid, the cissoid, and the solids generated by them. Then, the rectification of these curves and the area of the solids obtained from their rotation are considered. In the dedication to Robert Boyle, which introduces his treatise, Wallis states that with this paper he wants to go beyond the results presented by others on the subject, reaching universal conclusions by means of an easy-to-apply and spontaneous method [9] (p. 491), which is that of the Arithmetica infinitorum (The Arithmetic of Infinitesimals), and solving problems by means of suitably constructed numerical series.

This work by Wallis, albeit deeply studied by his contemporaries, has been neglected and almost forgotten with the advent of calculus. However, practical applications of the rectification of the parabola are still fundamental in some parts of science, such as ballistics [10], optics, and astronomy [11]. Moreover, the constructions of the parabola proposed by Wallis have an intrinsic educational value as shown in [12]. In fact, Wallis’ elaborations about the parabola are not limited to the problems of rectification reported in the present work. He made other relevant contributions to the study of conics. For instance, he showed in Proposition 12 (pp. 309–312) of the De Sectionibus Conicis Tractatus (Treatise on the Conics Sections) of 1655 [9] (pp. 291–354) that for every parabola built on a plane it is possible to construct the cone of which it is a section. This poorly known construction is used in [12] to propose an educational path in which the close relationship between conics as loci of points on a plane and conics as sections of a cone is highlighted.

This article is organized as follows. Section 2 tries to characterize the context in which Wallis’ intervention on the rectification of curves is inserted. Section 3 presents a concise introduction to Wallis’ Arithmetica method to better understand the meaning of what he deals with in the two small treatises. In Section 4, reference is made to the results of H. van Heuraet, W. Neile, and C. Wren that Wallis presents in his treatise. Section 5 contains the solution to the problem of the rectification of the parabola given by Wallis. Section 6 describes the passage that the author takes from rectification to flattening. Finally, some conclusions are drawn in Section 7.

2. The Context of the Wallisian Treatment

During the 17th century, the history of a particular curve, the cycloid, developed by reaching a peak in the set of problems that Blaise Pascal posed to other mathematicians in 1658, challenging them to solve those problems and reserving a prize for the winner. Throughout this history, we come across the issue of the primacy contested by Torricelli and Roberval on their solution to various problems to do with this curve and other problems posed by Pascal.

Pascal was astonished by the fact that the cycloid had not been studied by the Ancients, given its relevance to everyday life. He would have liked it to have been studied at least on a par with conics, of which Pascal himself was a keen student. This being the case, he issued a challenge, the object of which was to compute the center of gravity of the solids generated by the rotation around the axis and the base of the cycloid and their halves. He then claimed that some of the problems having to do with this curve had been solved by Roberval.

In the first part of his treatise, Wallis presents the history of this issue and gives his solution to the problems posed as well as other problems deriving from the study of this curve [9] (pp. 492–520). He was aware of Pascal’s 1658 work when writing these pages.

When elaborating the solution to these problems, he followed a method of his own, formulated in the Arithmetica infinitorum of 1656, that he had developed by reworking Cavalieri’s indivisible method, known through the writings of E. Torricelli, in an algebraic key [13,14,15,16].

We are unable to establish whether Wallis’ written text in this instance is the same as the one used in his answer to the challenge, but we can assume that this piece of writing is the reply to Pascal’s observations on the work that Wallis sent him.

This glance at the most contested issues concerning the cycloid in the years 1640–1660 leads us also to consider the context in which Wallis contributed to the debate.

In 1637, Mersenne and Roberval discussed the rectification of the arcs of a curve, comparing the arc of a cycloid with that of a parabola [17] (p. 118). Marin Mersenne was one of the people who played a key role in the history of the cycloid, as can be seen from his correspondence [17] and [18] (pp. 27–30).

In the same year, there was a rumor that Roberval had solved the outstanding problems concerning the curve starting from the idea of conceiving the curve as the “motion of a point”. During the winter of 1642, Thomas Hobbes was in Paris, probably in the company of the members of Mersenne’s circle, and took part in a discussion on the rectification of the spiral and the quadratrix and their relation to the parabola. He later discussed this in depth with Roberval. Some historical information in this regard can be found at the beginning of Pascal’s treatise entitled Dimension des lignes courbes (Dimension of curve lines) [19] (pp. 313–343).

Much has been written about this; however, we only have the letters of Mersenne, the writings of Roberval in On the length of the trocoide (cycloid), and the brief references made by Hobbes [15] (pp. 244–290) and [20] in Chapter 18 of Elements of Philosophy as hard evidence.

In 1658, Pascal published Dimensions des lignes courbes, in which he dealt with the problem of rectification by contrasting different curves. He returned to the same question in a letter to Amos Dettonville [19] (pp. 313–314). He inscribed and circumscribed polygons in the spirals and the parabola, considering only their perimeter [19] (pp. 320–322), and defined a correspondence between the arc of the parabola or spiral and a segment. The demonstration followed a traditional method, one often used by Archimedes. From this, we can infer Pascal’s conviction that it was not possible to arrive at the rectification of the parabola and spiral directly. We obtain the same impression from the pages of Dimensions des lignes courbes, in which he demonstrated that the cycloid had the length of a certain ellipse and challenged Wren’s conclusion (the point of departure of Wallis’ reflections) that the cycloid was equal to a straight line [19] (pp. 335–340).

This was the state of play when Wallis intervened into the discussion.

3. Wallis’ Method within the History of the Transmission of the Works of Archimedes

Before proceeding further, it is necessary to focus attention on what Wallis calls “his” method, which is formalized in the Arithmetica infinitorum of 1656.

It is known that, of all the works of Archimedes, Arab mathematicians of the 9th to 13th centuries were most familiar with the Measurement of a circle and On the Sphere and Cylinder. In the same period, other works were attributed to him, such as the short tract on the construction of the heptagon in a circle, no trace of which can be found in the works of Archimedes known in the West. The mathematicians of that period made use of methods that, in the light of later discoveries of Archimedes’ work, can be classified as “in the Archimedean style”, being easily comparable to the mechanical or the exhaustion method found in the Quadrature of the Parabola.

In the Latin tradition (from the Medieval period to the beginning of the 1600s), further writings of Archimedes were recovered and translated, but not the Method; that was found much later.

Thus, the mathematicians of the 17th century could study these texts directly; at the same time, they learnt about the Archimedean-style methods adopted by Arab mathematicians. We are not, however, in a position to delineate with any certainty how this transmission came about.

Reading Archimedes’ treatises did not lead them to simply re-propose the demonstrative and resolving methods learned; instead, they felt the need to give them an “updated” interpretation using the mathematical tools of the period.

A pivotal role in all this was played by the 1635 work of B. Cavalieri, Geometria indivisibilium (Geometry of Indivisibles), in which every geometric figure, be it rectilinear or curvilinear, was measured with the indivisibles. Echoes of the work of Cavalieri can be found in the Opere Geometriche (Geometric Works) of E. Torricelli (1644), which in turn provided a link to the diffusion of the “method of indivisibles” in Europe.

In the period 1650–1651, Torricelli’s writings became familiar to Wallis. In fact, it was these writings that led him to articulate his own method presented in the Arithmetica infinitorum.

The method he presented was of a numerical nature, consisting of the ratio between numerical series, passing from natural numbers to rational and irrational ones and their respective powers and roots; in this way, the argument extended to the relation between series with algebraic elements.

Each series begins with zero, while every partial ratio is as follows: the numerator is composed of the sum of n numbers beginning with zero and the denominator is the sum of n numbers all equal to the highest number of the numerator.

With the increase in the number of elements to be added, we realize the numerical value of the ratio. Hence, if n is the exponent of the elements to be added, the ratio between the sums of the two series, constructed as described above, tends to $\frac{1}{1+n}$.

In order to take this fully on board, let us consider a numerical series in which the elements are formed by the squares of natural numbers starting with zero. The ratios between the partial sums of the series and the partial sums of the terms equal to the highest term of the numerator are constructed in this way [9] (pp. 373–374):

$$\frac{0+1}{1+1}=\frac{1}{2}=\frac{1}{3}+\frac{1}{6}, \frac{0+1+4}{4+4+4}=\frac{5}{12}=\frac{1}{3}+\frac{1}{12}, \frac{0+1+4+9}{9+9+9+9}=\frac{14}{36}=\frac{7}{18}=\frac{1}{3}+\frac{1}{18}$$

$$\frac{0+1+4+9+16}{16+16+16+16+16}=\frac{30}{80}=\frac{3}{8}=\frac{9}{24}=\frac{1}{3}+\frac{1}{24}$$

$$\frac{0+1+4+9+16+25}{25+25+25+25+25+25}=\frac{55}{150}=\frac{11}{30}=\frac{1}{3}+\frac{1}{30}, \dots$$

From these and the following results, we can see that the increase in the number of terms added to the result always tends to be more than one $\frac{1}{3}$, to be considered, hence, a limit of all the ratios between the two series. We also note that $3=2+1$, where 2 is the exponent of the terms of the series: $0^2, 1^2, 2^2, 3^2, 4^2,$ ….

This result can be easily extended to the case in which a series is formed by cubes, to the power of 4, etc., and the ratios are constructed as above. In such cases, the ratios between these series tend to $\frac{1}{3+1}=\frac{1}{4}$, $\frac{1}{4+1}=\frac{1}{5}$, ….

Once the truth of the obtained results with exponents 2 and 3 had been verified, Wallis extended them to the n-th power; thus, the ratios of the partial sums tend to $\frac{1}{1+n}$.

In the same manner, the ratios to do with the series of square roots, cubes, and so on tend to $\frac{1}{\frac{1}{2}+1}=\frac{1}{\frac{3}{2}}=\frac{2}{3}$, $\frac{1}{\frac{1}{3}+1}=\frac{1}{\frac{4}{3}}=\frac{3}{4}$, etc. [9] (pp. 398–390).

To each of the studied series, Wallis associated a certain number of geometric problems that could be resolved by means of it.

The writing of Wallis is by no means easy to understand and interpret. It is certain, however, that it was studied by the young Isaac Newton and supplied the necessary groundwork for the study of infinite series.

Wallis frequently put forward geometrical or mechanical examples to be resolved with “his” method. Among these, we find an enquiry into the problem of the rectification of curves presented in the short treatises considered in this work.

When the work on the infinitesimal calculus began to be taken seriously, Wallis realized that his method was useless. Despite this, he proposed it again in the work Algebra in 1683, linking it with the method of exhaustion of Archimedes and Cavalieri’s indivisible method, attempting thereby to give it a certain respectability.

This brief introduction to Wallis’ method in the Arithmetica is indispensable if we are to grasp the significance of the contents of the two short treatises, which are presented here as a single work divided into two parts. Even though the ratios between series found above are not directly used in the proofs reported below, Wallis actually used them in the proofs he gave of some of the propositions (for instance at [9] (pp. 545–550)) contained in the tract. Indeed, certain passages dealing with Wallis’ discourse on mathematics are quite hard to fully grasp. The details of the calculations reported above have been given in order to help the reader who wants to read Wallis’ original work, as well as the parts and proofs not included in the present manuscript.

4. Wallis Presents Some Results of H. van Heuraet, W. Neile, and C. Wren

Wallis tackled the problems of rectification and flattening by returning to the solutions of the problems of rectification proposed by Hendrik van Heuraet (that Wallis deduced from a letter from Huygens, as he clarified, and not from the text written by its author), by William Neile (on this point, Wallis is the only witness as no writings have been passed down), and by Christopher Wren.

The intervention of van Heuraet (1633–1660), written in the final months of 1658 or at the beginning of 1659 [21] (pp. 517–20) and [22], consisted in demonstrating that a curve had an equal length to that of a line. Wallis noted that this result had to be considered part of a “symphonic expression”, referring to the general interest shown by mathematicians in the issue. He preferred not to coin a Latin term, such as rectificatio, which is not found in the literature, and instead used the Greek terms ευθυνσις (euthunsis) and ευθυσμος (euthusmos) (with preference for the former), which indicated the act of rectification.

Van Heuraet proposed, as an example of the rectification of a curve, the rectification of the semicubic parabola, considering it as an arc.

In 1659, Pierre de Fermat (1601–1665) intervened to resolve the problem by circumscribing the arc for half of the tangents drawn from their extreme points. The title of the memorial of Fermat is De linearum curvarum cum lineis rectis comparatione dissertatio geometrica (A Geometrical Dissertation on the Comparison of Curved Lines with Straight Lines), which was published in 1660 in an appendix to the volume by Antoine de Lalouvère (1600–1664) entitled Veterum geometria promota in septem de cicloide libris (Geometry of the Ancients on the Cycloid Developed in Seven Books), which was published in Toulouse in 1660. This author was one of the first contributors in the history of the challenge to resolve the problems concerning cycloids posed by Pascal.

With regard to the work of van Heuraet, Wallis referred to his own contributions of a geometric nature in Propositions 35 and 41 in his 1655 treatise on conics and in the scholium to Proposition 38 of the Arithmetica. In particular, in Proposition 35 on the equation of the hyperbole he deduced the measurements of the diameters, the straight side, and the ordinates [9] (pp. 339–340), in Proposition 41 he found the relation between the diameters conjoined to the hyperbole [9] (pp. 347–348), and in the scholium on Proposition 38 he raised the possibility of finding: a “certain” rectilinear figure equal to any circle; a straight line “quasi” equal to a parabola (ordinary, cubic, etc.) [9] (p. 380); and a straight line “quasi” equal to a genuine spiral (not Archimedean) [9] (p. 381).

Wallis returned to these problems later on, preferring for the moment to detail the procedure of Neile (1637–1670), whose work he learnt through Wren [9] (pp. 551–552).

We know little of W. Neile’s short life. He studied at Oxford and was a member of Wadham College, where in 1652 he studied mathematics with John Wilkins and Seth Ward. In 1657, he was a law student at the Middle Temple in London. He was a member of the private council of King Charles II. In 1657, he rectified an arc of the cubic parabola. This result was communicated by himself in the month of July or August 1657 [9] (p. 551) to William Brouncker (1620–1684) and Christopher Wren, who were part of the Gresham College Society, which after a few years became the Royal Society [23]. Wallis published the result achieved in the paper on the cycloid and the cissoid. In 1663, Neile was selected to be a Fellow of the Royal Society and in 1666 he was selected to be a member of the council.

Wallis also presented another demonstration by Neile, transmitted to him through Brouncker in 1657, but not published earlier because he was waiting for it to be published by Neile himself [9] (pp. 552–553).

Let me now focus on the pages of Wallis in which, after determining the properties of the barycentre [9] (pp. 499–532), he considered the conclusions of Christopher Wren (1632–1723), which Wren gave him while Wallis was at work on the first draft of the treatise on the cycloid [9] (p. 532).

Wallis’ pages were used to disseminate Wren’s elaborations [9] (pp. 533–541).

We will outline Wren’s method and conclusions regarding the rectification of the cycloid in general terms.

As regards the curve, the problems raised were addressed with the primary cycloid, obtained by the motion of a point on the circumference that moves along a line [9] (p. 541). Only after this were the results so far achieved extended to the protracted and the contracted cycloid. The issues Wallis raised with Wren began with the tangent line to the primary cycloid [9] (pp. 533–34), considered as a fundamental element in the subsequent steps, leading to the central problem, which was entitled Ευθυσμος Curvae lineae Cycloidis primariae secundum methodum Antiquorum demonstratum (Straightening of the curve of the primary cycloid demonstrated according to the method of the Ancients). Here, we find the expression ευθυσμος (euthusmos) used to indicate the right direction, the straightening. This term is rarely used in the literature, as there was no agreed-upon term to indicate rectification. Later on, Wallis, as implied in the title of the tract, used the term ευθυνσις (euthunsis) with the same meaning; this, however, along with its variants, had the advantage of being widely used in the literature (from Plato and Aristotle to Aeschylus, Euripides, and Galan).

The work followed the method of the Ancients in all its parts, both as regards the role of the definitions and, above all, the method of exhaustion. The aim was to calculate the length of a curve, not to calculate an area or a volume; only the perimeters of polygons inscribed or circumscribed in a circle were considered, as this allowed the scholars to compare those with the length of a curve.

Wren’s argument, read through the writings of Wallis, began with a definition of sizes declining to infinity, thereby opening up the possibility of constructing ever-diminishing proportions [9] (p. 534).

He defined the “indented polygon” (the Latin term is “serratum”) as “a figure that alternates sides between their parallels and oblique sides” and called “sides” only the oblique sides [9] (p. 535). He claimed it is possible to inscribe, in a semicircle, an indented polygon, the sum of whose sides is less than double the diameter, and to circumscribe around the semi-circle another indented polygon, the sum of whose sides is greater than the double of the diameter, such that the difference between the sum of the sides is equal to a given size.

Given the arbitrary nature of the chosen size, with the increase in the sides of the indented polygon, this difference became ever greater.

The formulation and the solution to the problem were obtained fully respecting the methods of the Ancients (the method of exhaustion) [9] (pp. 535–536).

This result was the basic premise for the study of the cycloid.

We should underline the fact that, when Wallis wrote his treatise, it was already known that the area enclosed by a cycloid was three times that of the generating circle.

By comparing the cycloid with the generating circle, it was possible to construct a polygon inscribed with or circumscribing a semi-cycloid.

This problem was addressed by the following lemma. The construction related to this problem is shown in Figure 1.

Lemma 1.

If from any point$O$of a primary cycloid$SPOAD$a tangent$OV$is drawn towards the base, the parallels$NOB$and$VPC$are drawn intersecting the generating circle at$B$and$C$, $A$is joined with$B$and$C$, and$OE$is drawn parallel to$AC$, then the tangent$OV$is greater, while$OE$is less than the portion of the curve$OP$intersected by the parallel lines [9] (p. 537).

Figure 1. Construction related to Lemma 1.

Figure 1. Construction related to Lemma 1.

This brings us to the first significant result of the following theorem.

Theorem 1.

The primary cycloid is four times the diameter of the generating circle [9] (p. 537).

This result can be generalized in the following way:

Theorem 2.

The primary cycloid is four times the diameter and equal to a portion of the assigned line [9] (p. 538).

The problems concerning the primary cycloid are also found in the contracted cycloid and the protracted cycloid.

These results concluded the first part of the treatise on the cycloid.

Up to now, there has been no sign of the Arithmetica infinitorum.

5. A Letter from Wallis to Chr. Huygens Containing the Solution to the Problem of Rectification by Means of the Method of Arithmetica Infinitorum

The second part of the treatise is accompanied by a letter addressed to Christian Huygens (1629–1695), in reply to one sent by Huygens on 9 June 1659, in which Wallis dealt with the question of the relation between the cycloid and the ellipse [9] (pp. 542–569).

The letter starts by resuming with the question of the challenge regarding the problems concerning the cycloid, referring to the history of this curve and the writings of Pascal.

In tackling the problems linked to rectification [9] (pp. 544–545), Wallis recalls Pascal’s conclusive theorem and demonstration in the Dimension des lignes courbes, with which Pascal connected the parabola and the spiral following the methods of the Ancients.

Wallis had no intention of taking part in the polemic between those who followed the ideas of Roberval (Hobbes bore witness to this; see [24] (p. 6)) and others who supported Pascal’s argument but, as was often the case (the polemic with Hobbes is a good example), he could be quite trenchant on other points.

As we have seen, over the course of the 17th century, mathematicians were engaged in finding new methods and solving new problems. Each scholar tried to affirm at the same time the goodness of the method used and the novelty of the results achieved. In this context, it is appropriate to understand the positions of Roberval, Torricelli, Pascal, and Wallis and the claim made by each of them about the historical priority in the achievement of the results.

Wallis devoted 25 pages of the treatise to the rectification of curves, passing from the cycloid to the cissoid of Diocles and the conchoid of di Nicomedes, leading to a generalization covering any type of curve. Finally, he unveiled the method of the Arithmetica infinitorum, with which he solved the problems of rectification and flattening, and he used the concepts of center of gravity and momentum with respect to a straight line.

The noted similarity between the cycloid and the cissoid recalls a “highly elegant” proposition made by Huygens on the cissoids [9] (p. 545), whose construction is reported in Figure 2.

Proposition 1.

The cissoid$C{S}_i$is such that the line$S_i{Z}_i$, that tends to$C$, intersected by the cissoid and the circumference of the circle$C{Z}_i$, is cut into two parts by the line$G{H}_i$that is perpendicular to the diameter$FC$; thus, the area$FC{S}_{i}AF$, of infinite length, delimited by cissoid$C{S}_i$and tangent$FA$to the circle, is three times the semicircle$F{Z}_{i}C$.

Figure 2. The construction related to Proposition 1 by Huygens on the cissoid.

Figure 2. The construction related to Proposition 1 by Huygens on the cissoid.

Wallis replied to Huygens’ request to address, with the Arithmetica method, infinite hyperbolas and cissoidal space. Thus, in this work, in finding a solution to this proposition we encounter the first application of his method.

We have the impression that the initial results obtained by applying the Arithmetica method have the simple purpose of showing the effectiveness of the same method in studying problems of rectification and flattening.

Finally, Wallis presented his proof of rectification [9] (pp. 553–554).

In order to simplify Wallis’ demonstration in a language closer to our own, we have modified his notations, as reported in Figure 3, and made explicit all the calculations omitted, as sometimes it is difficult to follow his line of reasoning in this part.

Let $ADP$ be a parabola with side $l$.

Let the diameters $A{D}_i$ be in arithmetical proportion to each other, i.e., $A{D}_i=id$.

Let $D_i{D}_{i+1}=d$ be infinitesimally small.

Let us assume $AD=a$.

Consequently, the total number $N$ of segments in which $AD$ is divided will be $N=\frac{a}{d}$.

The ordinates applied to the diameter are $D_i{P}_i=\sqrt{lA{D}_i}$, that is, $D_i{P}_i=\sqrt{idl}$, in particular, $DP=\sqrt{al}$.

The area of the parabolas $A{D}_i{P}_i$ is:

$$A_{\left(A{D}_i{P}_i\right)}=\frac{2}{3}{D}_i{P}_i·A{D}_i=\frac{2}{3}\sqrt{idl}·id=\frac{2}{3}\sqrt{{\left(id\right)}^{3}l}$$

The infinite rectangles $D_i{D}_{i+1}{P}_{i+1}{P}_i$ have an area of:

$$A_{\left(D_i{D}_{i+1}{P}_{i+1}{P}_i\right)}=D_i{D}_{i+1}·D_i{P}_i=d\sqrt{idl}$$

The ratio between the areas of the rectangles $D_i{D}_{i+1}{P}_{i+1}{P}_i$ and the areas of the corresponding parabolas $A{D}_i{P}_i$ is:

$$\frac{A_{\left(D_i{D}_{i+1}{P}_{i+1}{P}_i\right)}}{A_{\left(A{D}_i{P}_i\right)}}=\frac{d\sqrt{idl}}{\frac{2}{3}\sqrt{{\left(id\right)}^{3}l}}=\frac{3}{2i}$$

Let us draw a curve $A{E}_{i}E$ that intersects the parabola at $P$, such that the segments $D_i{E}_i$ are in proportion to the areas of the parabolas $A{D}_i{P}_i$.

Since the areas of parabolas $A{D}_i{P}_i$ are in proportion to $\sqrt{{\left(id\right)}^3}$, the same will apply to $D_i{E}_i$.

We shall call this a “semicubical parabola” and indicate its diameter by $A∆$.

Being $D_i{E}_i$ in proportion to $A_{\left(A{D}_i{P}_{i+1}{P}_i\right)}$, we have:

$$D_i{E}_i=k·A_{\left(A{D}_i{P}_i\right)}=k·\frac{2}{3}id\sqrt{idl}$$

Given $Nd=a=AD$, for $i=N$ we obtain:

$$D_N{E}_N=DE=k·\frac{2}{3}Nd\sqrt{Ndl}=k·\frac{2}{3}a\sqrt{al}$$

Since $DE=D_N{P}_N=DP=\sqrt{Ndl}=\sqrt{al}$, by equating the two expressions we have:

$$k·\frac{2}{3}a\sqrt{al}=\sqrt{al}$$

from which $k=\frac{3}{2a}$.

Thus,

$$D_i{E}_i=\frac{3}{2a}·\frac{2}{3}id\sqrt{idl}=\frac{id}{a}\sqrt{idl}=\frac{\sqrt{l}}{a}\sqrt{{\left(id\right)}^3}$$

Being $D_i{E}_i$ proportional to $A_{\left(A{D}_i{P}_i\right)}$, the differences between these will be in proportion to the differences between the areas of the corresponding parabolas.

Thus,

$$F_i{E}_i=D_i{E}_i-D_{i-1}{E}_{i-1}=k\left[A_{\left(A{D}_i{P}_i\right)}-A_{\left(A{D}_{i-1}{P}_{i-1}\right)}\right]$$

In particular,

$$\begin{gathered} FE=k\left[\frac{2}{3}\sqrt{a^{3}l}-\frac{2}{3}\sqrt{{\left(a-d\right)}^{3}l}\right]=\frac{2}{3}k\sqrt{l}\left[\sqrt{a^3}-\sqrt{{\left(a-d\right)}^3}\right] \\ =\frac{2}{3}k\sqrt{a^{3}l}\left[1-\sqrt{{\left(1-\frac{d}{a}\right)}^3}\right] \end{gathered}$$

$d$ being infinitesimally small (or $N$ very large), we can write:

$$\sqrt{{\left(1-\frac{d}{a}\right)}^3}\approx 1-\frac{3}{2}\frac{d}{a}$$

and consequently

$$FE\approx \frac{2}{3}k\sqrt{a^{3}l}\left[1-1+\frac{3}{2}\frac{d}{a}\right]=k\frac{d}{a}\sqrt{a^{3}l}=\frac{3}{2a}\frac{d}{a}\sqrt{a^{3}l}=\frac{3d}{2a}\sqrt{al}$$

Reasoning in the same manner, we can write:

$$F_i{E}_i\approx \frac{3d}{2a}\sqrt{idl}=\sqrt{\frac{9d^{2}l}{4a^2}\left(id\right)}$$

Analogously with $D_i{P}_i=\sqrt{idl}$, we can consider $F_i{E}_i$ as the ordinates to the diameter of a parabola with a straight side $\frac{9d^{2}l}{4a^2}$.

This is the parabola $ADC$, having $D_i{C}_i$ as the ordinates applied to the diameter.

Therefore, $D_i{C}_i=F_i{E}_i$.

If we consider the arc of curve $E_i{E}_{i-1}$ to be rectilinear (since $d$ is infinitesimally small), we can write:

$$E_i{E}_{i-1}{}^2=F_i{E}_i{}^2+d^2\approx \frac{9d^{2}l}{4a^2}\left(id\right)+d^2$$

We identify on segments $D_i{P}_i$, on the same side of parabola $ADC$, points $B_i$ such that

$$D_i{B}_i=E_i{E}_{i-1}\approx \sqrt{\frac{9d^{2}l}{4a^2}\left(id\right)+d^2}$$

Then, $A{B}_{i}BD$ is a trunk of the parabola, in which the straight side is $\frac{9d^{2}l}{4a^2}$, the minimal ordinate applied is $A{B}_0=d$, and the vertex $V$ moves to $\frac{4a^2}{9l}$ with respect to $A$; hence, $AV=\frac{4a^2}{9l}$.

The value $\frac{4a^2}{9l}$ corresponds (with the opposite sign) to the value of $id$ for which $D_i{B}_i=0$ (in the case in which $id=-\frac{4a^2}{9l}$).

Since $d$ is infinitesimally small, the area of parabola $ADC$ can be written as the sum of all the infinite rectangles having segments $D_i{C}_i$ as the base and $d$ as the height (such rectangles, in fact, have one side equal to an infinite arc of the parabola, which, however, can be considered a segment of the same order of approximation with which we have assumed that the arcs of curve $E_i{E}_{i-1}$ are segments).

Analogously, the area of the trunk of parabola $ADB{B}_0$ can be written as the sum of all the rectangles having base $D_i{B}_i$ and height $d$.

Therefore, we will have

$$\frac{A_{\left(ADC\right)}}{A_{\left(ADB{B}_0\right)}}=\frac{{{\displaystyle \sum }}_{i=1}^N\left(D_i{C}_i·d\right)}{{{\displaystyle \sum }}_{i=1}^N\left(D_i{B}_i·d\right)}=\frac{{{\displaystyle \sum }}_{i=1}^N{D}_i{C}_i}{{{\displaystyle \sum }}_{i=1}^N{D}_i{B}_i}$$

Since $F_i{E}_i=D_i{C}_i$ and $E_i{E}_{i-1}=D_i{B}_i$, and as ${\displaystyle \sum }_{i=1}^N{F}_i{E}_i=DE$ and ${\displaystyle \sum }_{i=1}^N{E}_i{E}_{i-1}=A{E}_{i}E$, $A{E}_{i}E$ being the length of the semicubical parabola, we obtain $\frac{A_{\left(ADC\right)}}{A_{\left(ADB{B}_0\right)}}=\frac{DE}{A{E}_{i}E}$, from which

$$A{E}_{i}E=\frac{A_{\left(ADB{B}_0\right)}}{A_{\left(ADC\right)}}·DE$$

Wallis provided a numerical example himself: if we assume $AD=a=4$, $DE=\frac{16}{3}$, it follows that $AV=1$ and the parabola $A{E}_{i}E=\frac{10\sqrt{5}-2}{3}$.

While Wallis terminated his discourse at this point, we prefer to continue and render his calculations more explicit.

From $DP=DE=\sqrt{al}$, we can obtain $l=\frac{D{E}^2}{a}=\frac{{\left(\frac{16}{3}\right)}^2}{4}=\frac{64}{9}$ and thus $AV=\frac{4a^2}{9l}=1$.

Thus

$$\begin{gathered} A_{\left(ADB{B}_0\right)}=\frac{2}{3}DB·VD-\frac{2}{3}AV·A{B}_0=\frac{2}{3}DB·\left(AD+AV\right)-\frac{2}{3}AV·A{B}_0 \\ =\frac{2}{3}\left(\sqrt{\frac{9d^{2}l}{4a}+d^2}·\left(4+1\right)\right)-\frac{2}{3}\left(1·d\right) \\ =\frac{2}{3}d\left(5·\sqrt{\frac{9l}{4a}+1}-1\right)=\frac{2}{3}d\left(5·\sqrt{5}-1\right) \end{gathered}$$

where $DB$ comes from $D_i{B}_i$ for $i=N$ (recall that $Nd=a$) and $A{B}_0=d$.

Moreover, $A_{\left(ADC\right)}=\frac{2}{3}AD·DC=\frac{2}{3}·4·\sqrt{\frac{9d^{2}l}{4a}}=\frac{2}{3}d·8$, where $DC$ comes from $D_i{C}_i=F_i{E}_i$ for $i=N$.

Therefore, $A{E}_{i}E=\frac{A_{\left(ADB{B}_0\right)}}{A_{\left(ADC\right)}}·DE=\frac{\frac{2}{3}d\left(5·\sqrt{5}-1\right)}{\frac{2}{3}d·8}·\frac{16}{3}=\frac{10\sqrt{5}-2}{3}$.

In all three demonstrations (the first two coming from Neile via Wren and Brouncker and the third coming from Wallis) involving the semicubical parabola, the starting point of Neile’s work, classical methods were used for the parabola and certain indispensable conditions were imposed in order to construct the semicubical parabola.

Up to this point, Wallis had made little use of the method he employed in the Arithmetica Infinitorum. In the demonstrations, he referred constantly to the classical definition of the parabola, derived from Apollonius, with its basic elements, and, at the same time, Archimedean techniques to divide a segment into infinitesimally small parts.

6. From the Rectification of the Curves to the Flattening of the Surfaces

From this point on, Wallis linked rectification with flattening: once the length of a curve is known, it is squared and the area calculated.

Although the problem does not seem too difficult theoretically, in concrete terms the problem is to identify which curve corresponds to the area calculated.

This general problem of rectification and flattening is resolved by starting from the conics and the spirals, curves which played a crucial role at the time, being considered the prototype of all other curves ancient, recent, and still to come.

Wallis tackled this problem by relaunching the discourse on the rectification of the parabola and recalling briefly the scholium of Proposition 38 in the Arithmetica, in which the problem of finding a straight line “almost” equal to a parabola was raised (see Figure 4).

This is what he wrote in the scholium [9] (pp. 380–381).

Again, in order to improve the readability of Wallis’ pages, we use more modern language and notation in the following proof.

Consider the semi-parabola $A{O}_{i}O$, whose tangent on the vertex $A$ is $AT$ (Figure 4).

Divide $AT$ in equal parts on points $T_i$ such that $T_i{T}_{i+1}=a$.

From each point $T_i$, let us draw $T_i{O}_i$ perpendicular to $AT$ and parallel to $A{D}_i$.

From Proposition. 23 of the Arithmetica [9] (p. 374), $A_{\left(AOT\right)} : A_{\left(ATOD\right)}=1:3$.

Consequently, $A_{\left(AOD\right)} : A_{\left(ATOD\right)}=2 :3$.

$T_i{O}_i$ increase in the same way as the square numbers 1, 4, 9, 16…, while the differences $D_i{D}_{i+1}$ between two of these increase in the same way as the odd numbers 1, 3, 5, 7....

Segments $O_i{O}_{i+1}$, inscribed in the parabola, from Pythagoras’ theory, are equal to the square root of the sum of the square of $T_i{T}_{i+1}$ (where $T_i{T}_{i+1}=a$) and of the square of the difference $D_i{D}_{i+1}$ (that increases like odd numbers).

Consequently, segments $O_i{O}_{i+1}$ increase as $\sqrt{a^2+1}$, $\sqrt{a^2+9}$, $.$...

Since these segments are inscribed in the parabola, assuming that a is very small, their sum is extremely close to the length (measure) of the arc of the parabola.

Then, Wallis joined $O_i{O}_{i+1}$ and $D_i{D}_{i+1}$ to the ordinates $M_i{X}_i$ applied to the diameter of the hyperbole $K{X}_{i}X$ and the triangle $L\mathsf{\Lambda }N$ of Figure 5 [9] (pp. 554–555). In this way, he set up a correspondence between the length of the arc of the parabola, the area of the hyperbole, and the area of the triangle.

Let us now see how he proceeded.

Given the previous parabola, let us now consider a segment $LK$ proportional to $T_i{T}_{i+1}=a$.

From $L$, we trace the line perpendicular to $LK$.

On this, we make $L{M}_1$ proportional $\mathrm{to} A{D}_1$ in the same ratio as $LK$ is proportional to $a$.

On $LN,$ we place points $M_i$, with $i>1$, such that $M_i{M}_{i+1}=2L{M}_1$.

From this, it follows that $L{M}_2=L{M}_1+M_1{M}_2=3L{M}_1$, $L{M}_3=L{M}_2+M_2{M}_3=5L{M}_1$, …; that is, $L{M}_i$ increase as odd numbers in the same way as segments $D_i{D}_{i+1}$.

Draw segments $K{M}_i$, thereby obtaining rectangle triangles $KL{M}_i$.

Then, $K{M}_i=\sqrt{K{L}^2+L{M}_i{}^2}$ increases in proportion to $\sqrt{a^2+1}$, $\sqrt{a^2+9}$, $\sqrt{a^2+25}$… and hence they are proportional to $O_i{O}_{i+1}$.

From points $M_i$, let us draw segments $M_i{X}_i$ parallel to $LK$ such that $M_i{X}_i=K{M}_i$.

Even if not demonstrated explicitly by Wallis, it is not hard to verify that points $X_i$ belong to hyperbole $K{X}_{i}X$.

Let us now draw the bisector $L\mathsf{\Lambda }$ of angle $KLN$.

Let $L_i$ be the points of intersection of $L\mathsf{\Lambda }$ with $M_i{X}_i$.

Thus, $M_i{L}_i=L{M}_i$ insofar as it is the cathetus of isosceles rectangle triangles $L{M}_i{L}_i$.

Hence, between segments $M_i{X}_i$ and segments $M_i{L}_i$ the same proportion exists as the one between segments $O_i{O}_{i+1}$ and segments $D_i{D}_{i+1}$.

$a$ being very small, $O_i{O}_{i+1}$ approximate the arcs of the corresponding parabola and, as a result, the sum of $O_i{O}_{i+1}$ approximates the arc of the parabola $A{O}_{i}O$.

From the proportion above, we will have:

$$\frac{{{\displaystyle \sum }}_i{M}_i{X}_i}{{{\displaystyle \sum }}_i{M}_i{L}_i}=\frac{{{\displaystyle \sum }}_i{O}_i{O}_{i+1}}{{{\displaystyle \sum }}_i{D}_i{D}_{i+1}}=\frac{parabolic arc A{O}_{i}O}{AD}$$

By multiplying the numerator and the denominator by $M_i{M}_{i+1}$, which in their turn are very small, the term of the numerator tends to the area of the quadrilinear $LNXK$ (segments $M_i{X}_i$ “fill” the quadrilinear area to use Wallis’ expression), while the term of the denominator tends to the area of the triangle $LN\mathsf{\Lambda }$ (segments $M_i{L}_i$ “fill” the triangle).

Consequently,

$$\frac{parabolic arc A{O}_{i}O}{AD}=\frac{A_{\left(LNXK\right)}}{A_{\left(LN\mathsf{\Lambda }\right)}}$$

Thus, from the length of the parabola we can obtain the quadrature of the hyperbole, and vice versa.

Wallis considered this result significant because it tied the length of a curve to the quadrature of another curve.

The move from rectification to flattening, from ευθυνσις (euthunsis) to πλατυσμος (platusmos), is more evident when we pass from the geometry of the plane to that of space, when from the considered curve, of which we have obtained the rectification, we pass to the respective solids of rotation (especially solids generated by cones, whose elements are studied by means of the series already present in the Arithmetica infinitorum).

In this small treatise, the question of flattening was not fully developed as Wallis preferred to engage in the debate on rectification, a controversial issue at the time.

After presenting the rectification of the parabola and the semicubical parabola, Wallis extended his results to other curves [9] (p. 562), starting from those with a shape similar to that of the parabola but with a different equation (for example, the curve of equation $\frac{y^4}{x^5}=l$, with $l$ a constant), and thereafter to the spiral and to other more complex curves. Always proceeding in a similar way to the one presented here, he generalized his discourse by applying his method [9] (p. 564) of the Arithmetica infinitorum.

7. Conclusions

In the present work, some of the contributions made by John Wallis to the rectification of curves, taken from his two short tracts (1659) on the cycloid and cissoids, are shown. These two tracts can be considered as two parts of a unique treatment, in which Wallis showed how to use his method to obtain the properties of curves different from conics and where he connected the problems of quadrature, reported in the first part, with the problems of rectification, which are specific to the second part. Even though Wallis’ approach to the problems presented in this article seems to be mostly of a geometric type, the objective that he proposes in the tract is to apply and advance his method founded on algebraic considerations, as shown in Section 3, based on the ratio between numerical series.

The project of Wallis privileges algebra over geometry and creates a method based on this choice (the one described in the Arithmetica) in such a way as to find, through this method, new applications and solutions to the issues considered worth studying by the mathematical community of the time.

In the work of mathematicians in the 17th century, it was important to balance the methods of the Ancients and the new methods, show how this balance could be achieved in practice, and demonstrate the extent to which the new methodologies and techniques could have a life of their own vis-à-vis the old ones.

Wallis explicates such an intersection between the old and the new theories in his treatise and the Archimedean methodologies provide a background and a cornerstone for all later constructions.

Reviewing his contribution and comparing it with the work of contemporaries dealing with the same or similar issues (from Roberval to Pascal and van Heuraet), we can conclude that Wallis manages to arrive at results far beyond those previously achieved.

Wallis’ approach, though it was well known and appreciated by his contemporaries (Newton himself was quite diligent in studying the tracts by Wallis), was soon forgotten with the advent of the infinitesimal calculus. However, his contribution deserves particular attention for different reasons. It produces remarkable geometric results without using the advanced concepts required by the calculus, and it therefore has educational value in all those cases in which the infinitesimal calculus cannot be directly applied. Furthermore, the problems connected to the rectification of conics still have importance in disciplines where approximations of parabolic arcs (for instance in ballistics or in the constructions of lenses, as in optics and astronomy) are required for improving the research on or the manufacturing of particular objects.