Mengoli on ``Quasi Proportions''*

HISTORIA MATHEMATICA 24 (1997), 257–280 ARTICLE NO. HM962147

MA.ROSA MASSA

Centre d’Estudis d’Histo`ria de les Cie`ncies, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain

This paper aims to analyze the first three elementa of the Geometriae speciosae elementa (Bologna, 1659) of Pietro Mengoli (1625–1686), probably the most original pupil of Bonaven- tura Cavalieri (1598–1647). In this work, Mengoli develops a new method for the calculation of quadratures using a numerical theory called ‘‘quasi proportions.’’ He grounds quasi proportions in the theory of proportions as presented in the fifth book of Euclid’s Elements, to which he adds some original ideas: the ratio ‘‘quasi zero,’’ the ratio ‘‘quasi infinite,’’ and the ratio ‘‘quasi a number.’’ A detailed analysis of this theory demonstrates the originality of Pietro Mengoli’s work as regards both its content and his method of exposition.  1997 Academic Press Le but de ce papier est l’analyse des trois premiers elementa de la Geometriae speciosae elementa (Bologna, 1659) de Pietro Mengoli (1625–1686), probablement le disciple le plus original de Bonaventura Cavalieri (1598–1647). Dans cette oeuvre Mengoli de´veloppe une nouvelle me´thode pour calculer des quadratures en utilisant une theórie nume´rique nommeé ‘‘theórie des quasi-proportions.’’ Il fonde cette theórie des quasi-proportions sur la theórie des proportions du cinquie`me livre des Ele´ments d’Euclid a` laquelle il ajoute quelques ideés originales: proportions ‘‘quasi-nulles,’’ ‘‘quasi-infinies,’’ et ‘‘quasi un nombre.’’ Une analyse de´tailleé de cette theórie de´montre l’originalite´ du travail de Pietro Mengoli.  1997 Aca- demic Press

L’objectiu d’aquest article e´s analizar els tres primers elementa de la Geometriae speciosae elementa (Bolonya, 1659) de Pietro Mengoli (1625–1686), que fou possiblement el deixeble me´s original de Bonaventura Cavalieri (1598–1647). En aquesta obra Mengoli desenvolupa un nou me`tode per calcular quadratures utilitzant una teoria nume`rica anomenada de ‘‘quasi proporcions.’’ Mengoli fonamenta les quasi proporcions en la teoria de proporcions del llibre cinque` dels Elements d’Euclides, a la qual hi afegeix unes nocions originals: rao´ ‘‘quasi nulla,’’ ‘‘quasi inﬁnita,’’ i ‘‘quasi un nombre.’’ Una exhaustiva ana`lisi d’aquesta teoria demostra l’originalitat de l’obra de Pietro Mengoli tant pel que fa a la seva forma d’exposicio´ com pel que fa al seu contingut.  1997 Academic Press MSC 1991 subject classiﬁcations: 01A45, 40-03, 40A25. KEY WORDS: Mengoli, 17th century, proportion, limit.

THE BACKGROUND TO MENGOLI’S WORK Most 17th-century mathematicians worked on problems of quadrature. From 1600 to 1680, the tools they used gave way to varied versions of inﬁnitesimals and indivisibles.1 Bonaventura Cavalieri (1598–1647) was one of the ﬁrst to develop a new method involving indivisibles, at a moment when there were two clear prece-

* A ﬁrst version of this work was presented at the University Auto` noma of Barcelona on October 1, 1993 for the Master’s degree in the history of science. 1 Among the many studies on this subject, the following are particularly useful: [5–7; 20; 25; 26; 28–30; 43].

257 0315-0860/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved. 258 MA. ROSA MASSA HM 24 dents: the technique of ancient times, today called the method of exhaustion (due to Eudoxus and Archimedes), and the work of Johannes Kepler (1571–1630).2 Pietro Mengoli’s Geometriae speciosae elementa (Bologna, 1659) also contains a new method for the calculation of quadratures. At the beginning of this work, in a letter dedicated to D. Fernando Riario, Mengoli outlines the relationship between his method of quadratures and the methods known up to that point: Both geometries, the old one of Archimedes and the new one of Indivisibles of my tutor, Bonaventura Cavalieri, as well as Vie`te’s algebra, are regarded as pleasurable by the learned. Neither through the confusion nor the mixture of these, but through their perfect conjunction, a somewhat new form [of geometry will arise]—our own—which cannot displease anyone. [32, 2]3 As this makes clear, Mengoli, who knew the work of Archimedes and Cavalieri well, introduces a new element into his geometry, namely, Vie`te’s algebra speciosa, which he quotes constantly.4 Another of Mengoli’s sources is Euclid’s Elements, which he mentions throughout the book in passages such as ‘‘. . . and I believe that I am not taking anything from others, except from the first nine [books] of Euclid’s Elements’’ [32, 9]5 and ‘‘I am not taking anything from others, except for certain aspects of Euclid, in the fifth and sixth, which I quote in the margin of the passages where I use it’’ [32, 2].6 Indeed, throughout the Geometriae speciosae elementa, Mengoli repeatedly uses Euclid’s definitions and theorems, but it is in the elaboration of the new theory of ‘‘quasi proportions’’ in the Elementum tertium of his Geometria that his use of the Elements is clearest. He first defines the notion of quasi proportion and then proves that quasi proportions satisfy all of the properties of standard proportions as found in the fifth book of the Elements. Mengoli wants to establish his new theory by grounding it in the first principles of Euclid’s Elements. To what extent, however, does Mengoli’s new method of quadratures follow from Cavalieri’s theory of indivisibles? After all, Mengoli was Cavalieri’s pupil, so some commonality would seem reasonable. Surprisingly, a study of Mengoli’s work reveals that the basis of his method was the theory of quasi proportions, a numerical theory of summations of powers and limits of these summations which has nothing to do with Cavalieri’s Omnes lineae.7 It is not clear why Mengoli did not follow in the path of his master. Cavalieri’s

2 Cavalieri’s method is set forth in [16; 17]. Kepler’s work on quadratures and cubatures is found in [24]. 3 ‘‘Ipsae satis amabiles litterarum cultoribus visae sunt, utraque Geometria, Archimedis antiqua, & Indivisibilium nova Bonaventura Cavallerij Praeceptoris mei, necnon & Viettae Algebra: quarum, non ex confusione, aut mixtione, sed coniunctis perfectionibus, nova quaedam, & propria laboris nostri species, nemini poterit displicere.’’ 4 Mengoli may have known Vie`te’s algebra speciosa through the second volume of Herigone’s textbook [23], as he cited Herigone as a source for his notation. See [32, 12]. 5 ‘‘ideoque nihil alienum sumpsi, praeterquam ex prioribus novem Elementis Euclidis.’’ 6 ‘‘Nihil alienum sumo; praeter quaedam, ex Euclide, in quinto, & sexto: quae suis locis allego, in margine.’’ 7 Mengoli had already published a work, Novae quadraturae arithmeticae (Bologna, 1650), in which he worked with inﬁnite series, adding them together and giving them properties. See [3; 21]. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 259 method did receive considerable criticism, and Mengoli might have been affected by this. In the letter dedicated to Giandomenico Cassini8 in the Elementum sextum of the Geometriae, Mengoli explains that 11 years before he had found many quadratures of plane ﬁgures using Cavalieri’s method. He goes on to acknowledge that he did not make them known on account of the attacks leveled against that method:

Meanwhile I left aside this addition that I had made to the Geometry of indivisibles, because I was afraid of the authority of those who think false the hypothesis that the infinity of all the lines of a plane figure is the same as the plane figure; I did not publish it not because I agreed with them, but because I was doubtful of it, and I tried . . . to establish new and secure foundations for the same method of indivisibles or for other methods which were equivalent. [32, 364]9

In this letter, Mengoli recognizes that the basis of Cavalieri’s method of indivisibles was not sound enough, and, as he wanted to ground this method of indivisibles solidly, he started on a new path, that of infinite series. In fact, after 1650 through the influence of Vie`te and, above all, Descartes, algebraic methods became ever more accepted in the field of geometry, and interest in numerical work, such as interpolation and approximation, also increased. Other mathematicians of the period—such as Pierre de Fermat (1601–1665), Gilles Personne de Roberval (1602–1675), John Wallis (1616–1703), and Blaise Pascal (1623–1662) also used these methods.10 They aimed, among other things, to calculate the result which today would look like

1p ϩ ...ϩtp 1 lim ϭ , ͫ tpϩ1 ͬ pϩ1 for t tending to inﬁnity. This would have allowed them to square the parabolas y ϭ xp, for p any positive integer. Mengoli also calculates this using his theory of quasi proportions, and he does so in an original and general way in Theorem 42 of the Elementum tertium (see below).

PIETRO MENGOLI’S GEOMETRIAE SPECIOSAE ELEMENTA The name of Pietro Mengoli (1625–1686) appears in the register of the University of Bologna in the period 1648–1686. He studied with Bonaventura Cavalieri and

8 Giandomenico Cassini was a professor of astronomy at the University of Bologna from 1650 to 1669, before moving to Paris [9, 37]. 9 ‘‘Ipsam interim accessionem, quam Geometriae Indivisibilium feceram, praeterivi: veritus eorum authoritatem, qui falsum putant suppositum, omnes rectas figurae planae infinitas, ipsam esse figuram planam: non quasi hanc sequens partem; sed illam quasi non prorsus indubiam debitans: tentandi animo, si possem demum eandem indivisibilium methodum, aut aliam equivalentem novis, & indubijs prorsus constituere fundamentis.’’ 10 Information on these mathematicians may be found in the following sources: on Fermat, [26, 230]; Roberval, [5, 18–21; 43, 41–44]; Wallis, [44, 365–392; 38, 34]; and Pascal, [13, 240; 37, 171]. 260 MA. ROSA MASSA HM 24 ultimately succeeded him in the chair of mechanics.11 He graduated in philosophy in 1650 and three years later in canon and civil law. In his first period, he wrote two mathematical books, Novae quadraturae arithmeticae seu de additione fractionum (Bologna, 1650) and Geometriae speciosae elementa (Bologna, 1659). He took holy orders in 1660 and until his death was prior of the church of Santa Maria Maddalena in Bologna. Although he published nothing between 1660 and 1670, the latter year saw the appearance of three works: Refrattioni e parallase solare (Bologna, 1670), Speculationi di musica (Bologna, 1670), and Circolo (Bologna, 1672). These reflected Mengoli’s new aim of pursuing research not on pure but on mixed mathematics like astronomy, chronology, and music.12 Furthermore, his research was clearly in defense of the Catholic faith [8, 23]. Mengoli went on writing in this line, publishing Anno (Bologna, 1675) and Mese (Bologna, 1681) on the subject of cosmology and Biblical chronology and Arithmetica rationalis (Bologna, 1674) and Arithmetica realis (Bologna, 1675) on logic and metaphysics. The present study focuses on the Geometriae speciosae elementa, a 472-page text in pure mathematics composed of an introduction, entitled Lectori elementario, which provides an overview of the six, individually titled chapters—or Elementa— that follow. In the first Elementum, De potestatibus, a` radice binomia, et residua (pp. 1–19), Mengoli shows the first 10 powers of a binomial given with letters for both addition and subtraction, and says that it is possible to extend his result to higher powers. The second, De innumerabilibus numerosis progressionibus (pp. 20–94), contains calculations of numerous summations of powers and products of powers in Mengoli’s own notation, as well as demonstrations of some identities. In the third, De quasi proportionibus (pp. 95–147), he defines the ratios ‘‘quasi zero,’’ ‘‘quasi infinity,’’ and ‘‘quasi a number.’’ With these definitions, he constructs a theory of quasi proportions on the basis of the theory of proportions found in the fifth book of Euclid’s Elements. The fourth Elementum, De rationibus logarith- micis (pp. 148–200), provides a complete theory of logarithmical proportions from Euclid’s Elements, while in the fifth, De propriis rationum logarithmis (pp. 201–347), Mengoli constructs the logarithm and its properties using the previous results. Finally, the sixth Elementum, De innumerabilibus quadraturis (pp. 348–392), in- volves calculating the quadratures of curves that correspond to functions now represented by y ϭ xs и (1 Ϫ x)pϪs. Mengoli does this using the theory of quasi proportions explained in the Elementum tertium. He also calculates barycenters of the areas of these curves.

11 In the academic year, 1648–1649, he held the post of Ad Arithmeticam; in 1650–1651, he took the chair of Ad Mechanicas, which had been Cavalieri’s; and in 1678, he assumed the position of Ad Mathematicam, which he held until his death. The program of Ad Mechanicas only lasted a few years and addressed current topics: Legant librum aequae ponderis Archimedis, Legant mechanicas marchionis Guidubaldi a Monte, and Legant de centro gravitatis. The program of Ad Mathematicam was always the same: Euclide, la teoria dei planeti, l’astronomia di Tolomeo [4, 817]. For more biographical information on Mengoli, see [34, 303; 9, 1]. 12 In the ﬁrst pages of the Circolo, Mengoli explained that he had found this result, the quadrature of the circle, in 1660, but had not published it because, according to him, he only wanted to publish the mathematics he needed to explain natural events [33, 1]. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 261

Stylistically, the Geometriae is organized, systematic, and rigorous. As noted above, Mengoli opens the book with an introduction, in which he explains the contents of each Elementum without technical terms or demonstrations. In so doing, he helps the reader to understand his ideas.13 Next, every Elementum has the same structure: a dedication followed by some definitions and theorems with their demonstrations. The dedications take the form of letters, such as those to Riario and Cassini quoted previously, to important figures in the scientific and cultural world who had (or had had) some connection with Mengoli. In these dedicatory letters, Mengoli comments on how he obtains his ideas, the notions that form their basis, and their originality. As far as the definitions are concerned, he explains all the terms used, even the ones that were already known or in use. Prior to the first theorem of the Elementum primum and on a separate page under the title Explicationes quarundam notarum, Mengoli also explains the basic notations that he will use throughout the book—addition, subtraction, the equals sign, and ratio— and he names all the letters and algebraic expressions that his analysis will involve. Nevertheless, one of the main difficulties in understanding the book concerns the notation; it is original and becomes more complicated as the text progresses.14 Mengoli proves theorems starting from clear hypotheses and explicitly stated properties, showing everything necessary. Even though the demonstrations are for specific values, they can be adapted perfectly to any appropriate value without loss of validity. In his proofs, Mengoli writes Hypoth. after the declaration and specifies which hypothesis has served as a starting-point. He then writes Demonstr. and proceeds to a step-by-step demonstration. In the margin he notes the theorems used in each line.15 Indeed, the work bears many similarities to a modern book and shows that Mengoli was ahead of his time in treating his subject with a high degree of rigor.

SOME BASIC RESULTS In order to understand the construction of the theory of ‘‘quasi proportions,’’ consider the results Mengoli obtains and uses in the ﬁrst two Elementa. In the Elementum primum, he deﬁnes three triangular tables. One of these corresponds to the table of the combinatory numbers, also called the arithmetic triangle, which, like Bosmans and Cassina, we believe to have been used frequently in mathematics

13 Relative to the introduction of the Novae quadraturae, Giusti notes that ‘‘Mengoli uses for the first time a method of exposition which he will use repeatedly afterwards. It consists in preceding the treatise with a long introduction in which he briefly outlines the principal results which he will arrive at in the book’’ [21, 200]. 14 At the time Mengoli wrote, there was no unanimity regarding different alternative symbolisms. For more on this, see [27, 159]. On the originality of Mengoli’s notation, consult [41, 617]. 15 At times he also writes Praepar. before the demonstration to specify a particular supposition that is necessary for the demonstration. On other occasions, as in the last nine propositions, which he calls problems, he writes Constr., as Euclid did, before the demonstration and explains the construction used in it. 262 MA. ROSA MASSA HM 24 from the middle of the 16th century onwards.16 Mengoli had already mentioned the principle of his table in one earlier work, Via regia ad mathematicas per arithmeticam, algebram speciosam, and planimetriam, ornata maiestati serenissimae d. christinae reginae suecorum (Bologna, 1655) [31, 11], and in the introduction of the Geometria, he says that the analysts called it Tabula multiplicium [32, 16]. Later, in the Circolo (Bologna, 1672), when he mentions this table again, he cites Vie`te’s Angular Sections as his source.17 In reference to another of the tables, the Tabula proportionalium, Mengoli notes its similarity to the one shown in Euclid VII.2.18 Moreover, in the dedication of the Elementum primum [32, 2], he remarks that these triangular tables were found in the first lesson of Algebra speciosa. It is difficult for us to identify his source with any certainty, but we may assume that these tables were known by most mathematicians of the time. Therefore, Mengoli’s originality stems not from the definition of these tables but from his treatment of them. On the one hand, he uses these tables and Vie`te’s algebra to create other tables with letters expressing additions of powers and products of powers; on the other, he employs the relations between these summations and the combinatory numbers of the arithmetic triangle to prove one of the important results of his book, namely, the sum of the pth powers of the first t Ϫ 1 integers. Consider now Mengoli’s techniques for constructing these triangular tables. The first, the table ‘‘of proportionals’’ [proportionalium], presents numbers expressed by letters so that in every row the first two always have the same ratio a : r. They also have the same ratio in the diagonals, 1 : a and 1 : r, respectively, because the letter u (see Fig. 1a) placed in the vertex represents unity.19 So Mengoli orders the numbers in a continuous proportion and displays them in a way that makes their identification easy. The second table ‘‘of multiples’’ [multiplicium] is the triangular table of combinatory numbers already mentioned. The third triangular table, ‘‘of nouns’’ [nominum], is the result of the combination of the two previous tables, but Mengoli actually never uses it later.20 He does employ the first two tables to build up tables of summation in the Elementum secundum.21

16 On the diffusion of this triangle, see [12, 21–24; 14, 33]. This triangle has passed into history as Pascal’s triangle because he explained and demonstrated its properties in a very clear style. See [37, 91–107; 12, 25–36]. Mengoli probably did not know of Pascal’s treatise since it was published in 1665, but he may well have known its source, Herigone’s work, as noted above. 17 Mengoli said: ‘‘Anzi vedasi con l’aggiunta delle unita` ne i lati, e in cima, come nel primo degli Elementi della mia Geometria Speciosa io la rappresento, e la definisco, e spiego ivi le sue proprieta`, e nel secondo, terzo, e sesto, l’uso ancora, e il Vietta, che ne fu` l’autore, nell’Algebra Speciosa, e nel suo Libro delle Settioni angolari’’ [33, 3]. In Vie`te’s work, I have found similar tables in [42, 295–299]. 18 I have not found this table in Euclid’s Elements, but there is a reference to a similar table in a 13th-century Latin edition of the Elements published by Johan Ludvig Heiberg and H. Menge in [12, 22; 14, 35]. 19 ‘‘Pro charactere autem unitatis, litteram u collocavimus, in vertice triangularis tabulae ‘‘[32, 13]. Also, when Mengoli uses the word ‘‘unitatem’’ in a statement, the letter u appears in the corresponding demonstration. Besides, without ever naming zero, either as a power or a number, he defines the ‘‘order’’ of u as one unit less than the first power. 20 The elements of this table are the development of the powers of the binomial a ϩ r or a Ϫ r, adding the corresponding signs depending on whether the binomial contains an addition or a subtraction. He demonstrates these developments in Theorems 8 and 10 of the first Elementum [32, 15]. 21 The tables are the same as in [32, 7]. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 263

FIGURE 1

There, Mengoli builds some summations in a very original way. For the benefit of his own calculus, he puts them in other triangular tables which he defines using the preceding ones. He considers an arbitrary number or tota, represented by the letter t, and divides it into two parts, a and r ϭ t Ϫ a. In his words, ‘‘The parts of tota will be called the separated part and the remaining part, and the separated part will be represented by the letter a and the remainder by r’’ [32, 21].22 He then takes tota equal to 1, 2, . . . and gives examples as far as 10. That is to say, if t is 2, a is 1, and r is 1. If t is 3, a may be 1 or 2, and r is then 2 or 1, respectively. If t is 4, a may be 1, 2, or 3, and r is then 3, 2, or 1, respectively, and so on. He also calculates the squares and cubes of a, the products of a and r, of the squares of a and r, etc. Moreover, in Definition 4, Mengoli explains that all the numbers a that he separates from the same number t (likewise the remainders r) will be called ‘‘syn- onyms’’ [synonymae]. So, if t is 3, the synonymae are 1 and 2; if t is 4, the synonymae are 1, 2, and 3, etc. He then proceeds to add the synonymae in order to obtain a summation of the type

tϪ1 O.a ϭ all the synonymae of t ϭ ͸ a. aϭ1 Mengoli calls this the mass of all the abscissae. So, if t is 3, the summation [massa] will be 3, because it is the sum of 1 and 2; if t is 4, the summation will be 6, because it is the sum of 1, 2, and 3, and so on.23

22 ‘‘2. Et partes Totae, dicentur, Abscissa, & Residua: & signiﬁcabitur abscissa, charactere a; & residua, r.’’ 23 As far as notation is concerned, Mengoli uses : to denote the equals sign, maior quam to denote Ͼ, and ; to denote ratio. Lower case letters express constants or given numbers, and capital letters denote variable quantities. Mengoli writes O.a, O.a2, O.r, putting, like Herigone, the exponents next to the letter. We will preserve the letter O to express the summations but not the notation of the power. Other symbols are in modern notation. 264 MA. ROSA MASSA HM 24

FIGURE 2

Mengoli orders all the summations resulting from the sum of all the synonymae in a table of proportionals [proportionalium]. He obtains a new triangular table ‘‘of symbols’’ [speciosa] (see Fig. 2). Its elements, O.u ϭ (t Ϫ 1) O.a ϭ 1 ϩ 2 ϩ 3 ϩ ...ϩ(tϪ1) O.a2 ϭ 12 ϩ 22 ϩ 32 ϩ ..ϩ(tϪ1)2 O.r ϭ (t Ϫ 1) ϩ (t Ϫ 2) ϩ (t Ϫ 3) ϩ .....ϩ1 O.ar ϭ [1. (t Ϫ 1)] ϩ [2. (t Ϫ 2)] ϩ ...... ϩ[(t Ϫ 1) .1], etc., are summations called species.24 Mengoli calls the first row of the triangular tables ‘‘of order one,’’ the second one ‘‘of order two,’’ and so on, and he assigns ordinal numbers to the rows or bases.25 He composes his table ‘‘of symbols’’ with the table of combinatory numbers or ‘‘of multiples’’ [multiplicium] to obtain the ‘‘subquadratrix’’ [subquadratrix] table (see Fig. 3). Its elements are subquadratrices. From this table, he builds yet a third by multiplying each row by a number one unit bigger than the order of the row (see Fig. 4). Mengoli calls this new table a ‘‘quadratrix’’ [quadratrix] table and its elements quadratrices. A seemingly strange construct, it may have been related to the calculus of lim [(1p ϩ ....ϩtp)/t(pϩ1)] ϭ 1/(p ϩ 1), for t tending to infinity. The denominator p ϩ 1 is the number by which each summation on the base p is multiplied. Mengoli may have known that the result could be applied to quadratures. Mengoli then proceeds to find and demonstrate the value of these quadratrices (summations) using the number t as the starting-point for their construction. These

24 He clearly takes the name from Vie`te and his Logistica speciosa, because he constantly refers to Vie`te and the analysts. 25 In the table of nouns, Mengoli calls the bases ﬁrst, second, ...,according to the order of the binomial. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 265

FIGURE 3

calculations give the value of the sum of the pth powers of t Ϫ 1 integers, a formula which was, in fact, not new. The first recognition of a general rule of this nature was apparently made in 1636 by Fermat, who announced that he had solved ‘‘what is perhaps the most beautiful problem of all arithmetic’’ [19, 69], namely, given any arithmetic progression, to find not only the sum of the squares or cubes but that of any power for all degrees to infinity. Fermat stated the rules but wrote neither the formula nor the demonstration.26 These rules are due to the fact that in the arithmetic triangle, the sum of all the figurate numbers of any order is expressible in terms of the figurate number of the next highest order.27 Eighteen years later, Pascal arrived, apparently independently, at a similar conclu-

FIGURE 4

26 On Fermat, see [19, 65–71 and 286–292; 26, 229–232; 13, 238–239; 12; 38–39]. Roberval [43, 171–173; 5, 18–21] and Wallis [38, 30–34; 44, 373–384] also expressed these summations for the first powers and thus deduced the value when the number of terms increases indefinitely. Roberval did not write the rules or the formulas for values of the power above 4; Wallis did not exceed 6. 27 Later, Bernoulli (1654–1705), in the Ars conjectandi (1713), deduced and wrote the general formula on the basis of these rules of polygonal numbers. See the third volume in [10, 164–168] and the translation from Latin in [39, 85–90]. 266 MA. ROSA MASSA HM 24 sion in the work, Potestatum numericarum summa (1654).28 Pascal proved the rules for four terms of a sum and for the power n ϭ 3 but argued that the proof held for all n. He enunciated the rules verbally and did not write the formula. Mengoli arrives at these results independently of Fermat and Pascal, by using Vie`te’s algebra to express the summations. Algebra allows him to obtain a certain level of generalization. Like Pascal and Fermat, he finds a rule in which the value of the sum of the pth powers is obtained only after determining the sum of the (p Ϫ 1)st powers, (p Ϫ 2)nd powers, etc. In addition to stating the rule, however, Mengoli also demonstrates it in Theorem 4 and uses it in Theorem 5 to perform 36 calculations. He closes with the statement: ‘‘And in infinity, it can be demonstrated, with the method shown above, every summation is equal to some totae’’ [32, 44].29 Mengoli puts these summations in a triangular table, so that he can obtain these indefinitely thanks to the general form he expresses in Theorem 22 (see below). The rule that Mengoli uses for calculating these summations is:

Theorem 4. Proposition 4. Any tota [raised to any exponent] is equal to the sum of the species formed by abscisses, raised to degrees smaller than the order of the tota and unity, multiplied by the numbers of the table of the ‘‘multiples’’ that correspond to the base whose order is equal to the power of the tota. [32, 36]30 In modern notation, this says:

aϭtϪ1 p aϭtϪ1 p tp ϭ a pϪ1 ϩиииϩ a0 ϩ1p. ͫ ͸ ͩ1ͪ ͬ ͫ͸ͩpͪ ͬ aϭ1 aϭ1 Mengoli bases his demonstration on the preceding theorems. In Theorem 1, he establishes the symmetry of the tables of summations and demonstrates the identity which today would be expressed as

aϭtϪ1 p aϭtϪ1 p (p ϩ 1) as и (t Ϫ a) pϪs ϭ (p ϩ 1) (t Ϫ a)s и apϪs. ͸ ͩsͪ ͸ ͩsͪ aϭ1 aϭ1 He then works out three examples: O.a3 ϭ O.r 3 ; O.a2r ϭ O.ar 2 ; and O.12a2r ϭ O.12ar 2. In Theorem 2, he ﬁnds two differences, which he calls incre- mentum. He proves for the elements of the lateral of the speciosa table (Fig. 2) that

aϭt aϭtϪ1 ͫ͸ a pͬ Ϫ ͫ ͸ a pͬ ϭ t p, aϭ1 aϭ1

28 On Pascal, see [37, 166–171; 13, 239–241; 12, 36–41; 7, 197]. 29 ‘‘Et in inﬁnitum, eadem methodo supra tradita, potest demonstrari, qualiter acceptis totis, quaeque massa est aequalis.’’ 30 ‘‘Tota quaelibet, est aequalis, aggregatis omnibus minu` s ordinatarum abscissarum speciebus, & unitati, acceptis secundum numeros multiplices, in base sibi aequeordinata iacentes.’’ HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 267 and, using the residua, shows that

aϭt aϭtϪ1 aϭtϪ1 p aϭtϪ1 p ((t ϩ 1) Ϫa) p Ϫ (t Ϫ a) p ϭ a pϪ1 ϩиииϩ a0 ϩ1p. ͫ͸ ͬ ͫ ͸ ͬ ͫ ͸ ͩ1ͪ ͬ ͫ͸ͩpͪ ͬ aϭ1 aϭ1 aϭ1 aϭ1

Since he has already proved that both incrementa are equal, Theorem 4 results, namely,

aϭtϪ1 p aϭtϪ1 p t p ϭ apϪ1 ϩиииϩ a0 ϩ1p. ͫ ͸ ͩ1ͪ ͬ ͫ͸ͩpͪ ͬ aϭ1 aϭ1

To get a sense of this, consider the example of t5 ϭ O.5a4 ϩ O.10a3 ϩ O.10a2 ϩ O.5a ϩ O.u ϩ u. Here, O. represents the summation from a ϭ 1toaϭtϪ1, and 5, 10, 10, 5 are the numbers of the ﬁfth base of the table of multiples (Fig. 1b). The lower powers of t are thus

t4 ϭ O.4a3 ϩ O.6a2 ϩ O.4a ϩ O.u ϩ u, t3 ϭ O.3a2 ϩ O.3a ϩ O.u ϩ u, and (1) t2 ϭ O.2a ϩ O.u ϩ u.

Using these expressions, Mengoli in Theorem 5 calculates

O.2a ϭ t2 Ϫ t, O.6a2 ϭ 2t3 Ϫ 3t2 ϩ t, O.4a3 ϭ t4 Ϫ 2t3 ϩ t2, up to 36 terms of the quadratrix table (Fig. 4). For example, he derives the value of O.4a3 from (1) as follows:

O.4a3 ϭ t4 Ϫ 2t3 ϩ t2. Demonstration 4.hO.4a3 ϩ O.6a2 ϩ O.4a ϩ O.u ϩ u ϭ t4 Sup. p O.u ϭ t Ϫ 1

Sup. 2 O.4a ϭ 2t2 Ϫ 2t Sup. 3 O.6a2 ϭ 2t3 Ϫ 3t2 ϩ t O.4a3 ϩ 2t3 Ϫ t2 ϭ t4 O.4a3 ϭ t4 Ϫ 2t3 ϩ t2. Quod & c. [32, 40]

Mengoli calculates O.6ar ϭ t3 Ϫ t, O.2772a5r 5 ϭ t11 Ϫ 22t5 ϩ 231t3 Ϫ 210t, using 268 MA. ROSA MASSA HM 24 similar incrementa for the elements in the middle of the table.31 He next generalizes these summations in the statement of Theorem 2232 Theorem 22. Proposition 22. Any quadratrix is equal to the tota raised one unit larger than the order of the base [where the quadratrix is found], minus the sum of other totae not raised above this order. [32, 74]33 In modern notation, this is equivalent to

aϭtϪ1 p (p ϩ 1) и as и (t Ϫ a)pϪs ϭ t pϩ1 Ϫ tn . ͸ ͩsͪ ͩ͸ ͪ aϭ1 nՅp Considering only the quadratrices of the lateral of the quadratrix table (Fig. 4), this says

aϭtϪ1 (p ϩ 1) ͸ ap ϭ t pϩ1 Ϫ ͩ͸ tnͪ. aϭ1 nՅp After demonstrating this theorem, Mengoli does not emphasize its applicability for calculating the value when t tends to inﬁnity (as Fermat and Pascal had done). In the Elementum tertium, however, he uses his result to ﬁnd this value. Mengoli continues performing similar calculations for (t Ϫ 1) and (t ϩ 1), reaching 37 theorems in total.

MENGOLI’S QUASI PROPORTIONS After performing all of these calculations in the first two Elementa, Mengoli elaborates the theory of quasi proportions in the Elementum tertium. There, in the dedicatory letter to D. Fabio Alamandino, he characterizes quasi proportions as ‘‘an as yet unknown geometric element which I have established to solve theorems, and most difficult ones, by means of an easy procedure’’ [32, 95].34 Mengoli also includes some surprising definitions seemingly unrelated to what he has calculated so far. In order to understand these definitions, it is necessary first to analyze the Mengolian meaning of some new terms that appear in them. The first definition begins with the words ‘‘Ratio indeterminata determinabilis,’’ but how did 17th-century mathematicians understand the phrase ‘‘determinable indeterminate ratio’’? Mengoli, in the introduction to the Elementum tertium, wants to clarify this notion: When I write O.a, immediately after the preceding chapter you have the mass of all the abscissae: but what value this mass is you do not yet know if I do not write of which number

31 For similar identities, see [2, 315]. 32 According to Mengoli, this theorem can be demonstrated by induction from the calculus of quadratrices (summations). He makes another demonstration apart from this one [32, 74]. 33 ‘‘Quaelibet quadratrix est aequalis totae unitate plus ordinatae, demptis, additisque aliqualiter acceptis totis, non plus ordinatis, qua`m sit eius basis.’’ 34 ‘‘inauditum hucusque Geometricum elementum, ad theoremata, caeteroqui difﬁcillima, facili negotio soluenda, cum instituerim.’’ HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 269

the mass is. But if I assign O.a to the mass of the number t, you do not know either how much it is if at the same time I do not assign the value of the letter t. But when I allow you to ﬁx a value for the letter t, and you, using this licence, say that t is equal to 5, immediately you will accurately assign O.a equal to 10, t2 equal to 25, t3 equal to 125, and O.r equal to 10, and if the letters t are determinate, the quantities O.a, O.r, t2, t3 will be determinate. Thus, before you have used the licence given, you actually had O.a, O.r, t2, t3, [which are] determinable [but] indeterminate quantities. [32, 61]35 Thus, it is clear that the summations are indeterminate numbers, but they are determinate when we know the value of t. By assigning different values to t, Mengoli introduces the concept of ‘‘variable,’’ a notion that was rather new at the time. There also seems to be a certain dependence between the value of t and the value of the summation, yet while Mengoli implies an idea of succession, his usage is still far from the general concept of function. Mengoli continues: But if I want to know the ratio of the mass O.a, of any number t,tot2; or the ratio of the mass O.2a to t2;orofO.ato t;orofO.ato t3, about which I surely have doubts, . . . when I assign values to t, then you know determinate ratios, but not always the same answers to the same questions. If the value 3 is given to the letter t, for the ratio O.a : t2 you will answer, 3 to 9; if you give the value 4 to the letter t, you will answer the same question by 6 to 16, which is not the same ratio as 3 to 9. We would answer with other values for other ratios. And so, given the licence for assigning values to the letter t, before assigning them, we have that the ratio O.a to t2 is indeterminate [but] determinable. [32, 62]36 Therefore, the value of the ratio is also indeterminate but is determinable by increasing the value of t. The ratio does not really assume this value, which we may interpret as its actual value; rather, it tends to it as t increases. It is in this sense that Mengoli understands the word ‘‘determinable.’’ Mengoli proceeds to give examples and to clarify his notion of ‘‘ratio quasi a number.’’ He considers values up to 10. For the ratio O.a to t2, he argues that it is nearer to 1/2 than any other ratio and so calls it ratio quasi 1/2: . . . for different values of the letter t, ordered always in an increasing [sequence] there are different [ratios] and always ordered in increasing [sequence] but always smaller than the ratio

35 ‘‘Cum scripsero O.a, statim ex praecedenti capite habes massam ex omnibus abscissis: sed quota sit haec massa, nondum habes, nisi scripsero, cuius numeri sit massa. Quod si assignavero O.a, numeri t massam esse; neque sic habes, quota sit, nisi simul assignavero, quotus est numerus, valor litterae t . . . Cum vero` licentiam dedero, ut quotum quemque litterae t valorem taxes; tuque huiusmodi usus licentia dixeris, t valere quinario: statim profecto assignabis & O.a, valere 10; & t2, valere 25; & t3, valere 125; & O.r, valere 10; & determinatae litterae t, determinatas esse quantitates O.a, O.r, t2,t3.Quare data licentia antequam usus fueris, habebas profecto O.a, O.r, t2,t3,quantitates indeterminatas determi- nabiles.’’ 36 ‘‘Sed si quaesiero, quaenam sit ratio Massae O.a, cuiuspiam numeri t,adt2; aut Massae O.2a,ad t2; aut Massae O.a,adt; aut Massae O.a ad t3 : ad has profecto` interrogationes, data licentia usus, cum taxaveris litterae t valorem, tunc determinatam assignabis rationem; sed non eamdem semper, ad eamdem quaestionem. Siquidem litteram t, taxaveris valere 3; pro ratione O.a,adt2, respondebis, 3 ad 9: qui si taxaveris litteram t valere 4; ad eamdem quaestionem respondebis 6 ad 16: quae non est eadem ratio 3 ad 9: item pro alijs valoribus, aliam respondebis rationem. Itaque data licentia taxandi litteram t, antequam taxaveris, habes rationem O.a ad t2 indeterminatam determinabilem.’’ 270 MA. ROSA MASSA HM 24

1/2; indeed approaching always nearer to the same 1/2. That is, if the question could be propounded for any value given I would answer that the ratio gets nearer to 1/2 than any other ratio [given], [and] it will be called to the same indeterminate ratio O.a to t2, quasi 1/2. [32, 62]37

Thus, the ratio takes different values as the value of t increases. Moreover, these values are nearer to 1/2 than any other given ratio. The difference between 1/2 and the ratio, which is determined when the value of t increases, is thus smaller than the difference between 1/2 and any other given ratio.38 The ‘‘limit’’ of this succession of ratios or of this ratio, as far as it is thus determinable, is 1/2, and Mengoli terms this ‘‘limit’’ ratio quasi 1/2. The idea of ‘‘ratio quasi a number’’ suggests, although in an imprecise way, the modern concept of limit.39 Mengoli also uses two other phrases in the definitions he presents in the Ele- mentum tertium which require special analysis and explanation. The first, ‘‘Quatenus ita determinabilis,’’ is translated here as ‘‘as far as it is thus determinable’’ in the sense of ‘‘tending to, as it takes on greater values.’’40 The second, ‘‘Ratio quasi aequalitas,’’ reflects Mengoli’s reliance on the Euclidean concept of positive ratio. The aequalitas of a ratio is thus nothing other than the equality of its terms, that is, unity. Mengoli uses the term in this sense throughout his book. The inaequalitas of a ratio denotes a number other than unity, and so ratios minor inaequalitas and maior inaequalitas correspond to numbers smaller and larger than unity, respectively. Given these interpretations, Mengoli makes the following definitions in the Elementum tertium:

1. A determinable indeterminate ratio, which, when determined, can be greater than any given ratio, as far as it is thus determinable, will be called quasi inﬁnite. 2. And one that can be smaller than any given ratio, as far as it is thus determinable, will be called quasi null. 3. And one that can be smaller than any given ratio greater than equality, and greater than any given ratio smaller than equality, as far as it is thus determinable, will be called quasi equality. Or otherwise, that which can be nearer to equality than any given ratio not equal to equality, as far as it is thus determinable, will be called quasi equality. 4. And one that can be smaller than any ratio larger than a given ratio, and larger than any ratio smaller than the same given ratio, as far as it is thus determinable, will be called

37 ‘‘pro varijs litterae t valoribus, ordinatim semper maioribus; varias, & semper ordinatim maiores esse: dimidia quidem ratione semper minores; ad ipsam vero` dimidiam semper propiu` s accedentes. Quod si propositae quaestioni potuerit, pro quodam valore assignabili responderi ratio propior dimidiae, qua`m alia quaelibet; dicetur ipsa indeterminata ratio O.a ad t2, quasi dimidia.’’ 38 Mengoli considers the given ratio as always different from 1/2. 39 In his Circolo of 1672, Mengoli again uses quasi ratios and explains: ‘‘Dissi quasi, e volsi dire, che vadino accostandosi ad essere precisamente tali’’ [33, 49]. 40 Note that Agostini translates the deﬁnitions but leaves out the sentence quatenus ita determinabilis [1, 21]. Cassina does translate it and interprets this sentence referring to the ﬁeld of applicability of a function f and to the values lower than the limit [15, 92]. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 271

quasi equal to this given ratio. Otherwise one that can be nearer to any given ratio than any other ratio not equal to it, as far as it is thus determinable, will be called quasi equal to the same [given] ratio. 5. And the terms of ratios quasi equal between them will be called quasi proportional. 6. And [the terms] of quasi equality ratios will be called quasi equal. [32, 97]41

The sixth and final definition, in light especially of the third definition, translates as ‘‘And the terms of ratios that are nearer to equality than any other given ratio other than equality, as far as these ratios are determinable, will be called quasi equal’’ [32, 97].42 This differs from lemma I in the first book of Newton’s Principia only in that the latter speaks of a certain finite time whereas Mengoli says: ‘‘as far as these ratios are determinable.’’ Both definitions express substantially the same idea and represent the first attempts to give an actual definition of the limit concept. Following these six definitions, Mengoli presents 61 theorems which can be di- vided into two broad groups on the basis of their content. He demonstrates the properties of quasi proportions in theorems 1 to 33. In the first six theorems, where there are no ‘‘quasi’’ expressions, Mengoli shows that the properties satisfied by the proportions in Euclid’s Elements are also satisfied when the greater than or less than signs are used instead of the equals sign. Mengoli uses the same names as Euclid and with the same meanings. He then explores, in Theorems 7 to 33, the properties of and relations between the new quasi expressions based on the six definitions and the first six theorems. Just as Theorems 1 through 6 prove that the properties—convertendo, componendo, per conversionem rationis, etc.—that Euclid verified for proportions hold for inequalities, Theorems 7 through 33 establish that they still hold even when the ratio is very large (quasi infinite) or very small (quasi null) or near a given ratio (quasi number). In order to comprehend better Mengoli’s manipulation of these ideas, consider the proofs of Theorems 4 and 8. The proof of Theorem 4 gives a sense of how Mengoli verifies properties of proportions when the sign of the proportion is greater

41 ‘‘1. Ratio indeterminata determinabilis, quae in determinari, potest esse maior, quam data, quaelibet, quatenus ita determinabilis, dicetur, Quasi infinita. 2. Et quae potest esse minor, qua`m data quaelibet, quatenus ita determinabilis, dicetur, Quasi nulla. 3. Et quae potest esse minor, qua`m data quaelibet maior inaequalitas; & maior, qua`m data quaelibet minor inaequalitas, quatenus ita determinabilis, dicetur, Quasi aequalitas. Vel aliter, quae potest esse propior aequalitati, qua`m data quaelibet non aequalitas, quatenus talis, dicetur, Quasi aequalitas. 4. Et quae potest esse minor, qua`m data quaelibet maior, proposita quadam ratione; & maior, qua`m data quaelibet minor, propositaˆ eaˆdem ratione, quatenus ita determinabilis, dicetur, Quasi eadem ratio. Vel aliter, quae potest esse propior cuidam propositae rationi, qua`m data quaelibet alia non eadem, quatenus talis, dicetur, Quasi eadem. 5. Et rationum quasi earundem inter se, termini dicentur, Quasi proportionales. 6. Et quasi aequalitatum, dicentur, Quasi aequales.’’ 42 The Latin text of Newton’s lemma is: ‘‘Quantitates, ut Et quantitatum rationes, quae ad aequalitatem tempore quovis finito constanter tendunt, Et ante finem temporis illius propius ad invicem accedunt quam pro data quovis differentia, fiunt ultimo aequales’’ [35, 28]. It is translated as follows: ‘‘Quantities, and the ratio of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal’’ [36, 29]. 272 MA. ROSA MASSA HM 24 than or less than, while that of Theorem 8 illustrates how Mengoli verifies the property componendo when the ratio is very large:

Theorem 4. Proposition 4. From larger ratios, by composition, the compound ratio will also be larger; and if ratios are smaller, it [will be] smaller. [32, 100]43

Hypothesis. a : b Ͼ c : d. e : f Ͼ g : h. I said a : b, ϩe : f Ͼ c : d, ϩg : h.

Preparation. a : b ϭ i : d. e : f ϭ d : l. g : h ϭ d : m.

Demonstration. constr. a : b ϭ i:d. hypoth. a : b Ͼ c : d. 13.5 i : d Ͼ c : d. 10.5 i Ͼ c. constr. e : f ϭ d : l. hypoth. e : f Ͼ g:h. 13.5 d : l Ͼ g:h. constr. g : h ϭ d : m. 13.5 d : l Ͼ d : m. 10.5 l Ͻ m. 8.5 i : l Ͼ c : m. p.p44 a : b, ϩe : f ϭ i : d, ϩd : l ϭ i : l. p.p. c : d, ϩg : h ϭ c : d, ϩd : m ϭ c : m. a : b, ϩe : f Ͼ c : d, ϩg : h. Quod & c. Quare & c. [32, 100]

Here, Mengoli’s use of the ﬁfth book of Euclid’s Elements is clear. First, from the hypothesis and the preparation, using Propositions 13 and 10 of the ﬁfth book of the Elements, he deduces that i Ͼ c and l Ͻ m. After this, using Proposition 8 of Book 5 twice, he obtains i : l Ͼ c : m. Finally, forming the product and simplifying, he arrives at a : b.e : f ϭ i : l Ͼ c : m ϭ c : d.g : h. Mengoli’s treatment of Theorem 8. Proposition 8 is also typical of his argumen- tation:

Ratio quasi infinite, componendo, is quasi infinite: also dividendo, is quasi infinite. [32, 103]45

43 ‘‘Ex maioribus rationibus, ex aequali, maior est ratio composita: & ex minoribus, minor.’’ Mengoli defines the term ex aequali in the Elementum primum: 14. A ratio is obtained ex aequali when it is obtained by composition of ratios. [32, 6]. 44 ‘‘p.p.’’ means the first proposition of the first element, where Mengoli demonstrates that the product of equal ratios is equal. 45 ‘‘Ratio quasi infinita, componendo, est quasi infinita: item dividendo est quasi infinita.’’ HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 273

Hypothesis. Given a quasi inﬁnite ratio A to B, I say that componendo a quasi inﬁnite ratio A ϩ B to B is obtained.

Preparation. We suppose that in any ratio c to d in which c is bigger than d, the difference will be e.

Demonstration. 8.5 If c is equal to or smaller than d, it is obvious that A ϩ B to B can be bigger than c to d. But, if c is bigger than d, def.1 asAto B can be bigger than e to d, p.h. then componendo, A ϩ B to B can be bigger than e ϩ d to d; prepar. but e ϩ d is c, consistently A ϩ B to B can be bigger than c to d.SoAϩBto def. 1 B is ratio quasi infinite. Quod & c. [32, 103] In this proof, Mengoli again relies on Euclid, using VIII.5 to go from A ϩ B Ͼ A to A ϩ B : B Ͼ A : B. This then allows him to argue, based on Definition 1, that since A : B Ͼ c : d, then A ϩ B : B Ͼ c : d. Following these preliminaries, Mengoli calculates the first quasi ratio in Theorem 34, a theorem which, at first glance, appears elementary, although he does not seem to think so. From this point on, the content of the Elementum tertium changes com- pletely.

Theorem 34. Proposition 34. The ratio of tota to unity is quasi inﬁnite.

Demonstration. As we do not say which number tota is, it is indeterminate, thus the ratio of tota to unity is also indeterminate. However, as this number is determinable, we could in fact say which number tota is, and from this the ratio of tota to unity is determinable. Finally, as we could say that this number tota is bigger than the ratio of a number to unity, which is any given ratio, this number, which is tota, will be the ratio of tota to unity and will be bigger than any given ratio. So, the ratio of tota to unity is quasi infinite. [32, 125]46 Mengoli supposes that since tota is indeterminate, he can always find a value of it bigger than a given ratio, and so the ratio of tota to unity (that is, tota itself) could be bigger than any given ratio. So, in light of Definition 1, this ratio is quasi infinite. In modern terms, if a positive number increases toward infinity, the ratio of this number to unity tends to infinity, or t lim ϭ ȍ. tǞȍ 1

46 Theor. 34. Prop. 34. Tota ad unitatem, quasi est inﬁnita. Demonstr. Nam tota, cum non dicatur, cuius numeri tota sit; est indeterminata: ideoque totae ad unitatem, ratio est indeterminata. Cumque possit dici, cuius numeri tota sit; est determinabilis: ideoque totae ad unitatem, ratio est determinabilis. Cum denique possit dici eius numeri tota, qui maior sit, qua`m vt ad unitatem, habeat quamlibet rationem datam; qui numerus, ipsa sui ipsius est tota: erit ratio totae ad unitatem, maior, qua`m data quaelibet. Ergo tota ad unitatem, quasi est inﬁnita.’’ 274 MA. ROSA MASSA HM 24

Theorem 34 is essential for the calculus of all quasi ratios because all other calculations are derived from it. From theorem 34, Mengoli calculates the quasis associated with a number t that is increasing. In a sense, he calculates what the ratios of polynomials tend toward, when the base gets bigger, depending on the degree of the numerator and of the denominator. To be precise, Mengoli proves the following theorems:

Theorem 35. The ratios (t Ϫ 1) : 1 and (t ϩ 1) : 1 are quasi inﬁnite. [32, 126]

Theorem 36. The ratios t :(t Ϫ 1), t :(t ϩ 1), (t Ϫ 1) : (t ϩ 1) are quasi equal. [32, 126]

Theorem 37. The ratios tm :1,(t Ϫ 1)m :1,(t ϩ 1)m : 1 are quasi inﬁnite. [32, 127]

Theorem 38. With m Ͼ n, the ratios tm : t n,(tϪ1)m :(t Ϫ 1)n,(tϩ1)m :(t ϩ 1)n are quasi inﬁnite. [32, 127]

Theorem 39. The ratios tm :(t Ϫ 1)m, tm :(t ϩ 1)m,(tϪ1)m :(t ϩ 1)m are quasi equal. [32, 128]

Theorem 40. With m Ͼ n, the ratio tm : ͸ tn is quasi infinite. [32, 128] nϽm

In Theorem 41, Mengoli, like Roberval [5, 19; 43, 172] and Wallis [38, 32; 44, 382–384] says that smaller powers could be ignored as t increases. He proves this from the properties of the quasi ratios alone.

Theorem 41. Proposition 41. The tota raised to the largest exponent and the same tota added to other totae raised to the lesser exponents, or subtracted, are quasi equal. [32, 129]47

Hypothesis. The tota raised to the largest exponent is A: the other totae added raised to the lesser exponents are B and the subtracted totae C. I say A, A ϩ B, A Ϫ C, A ϩ B Ϫ C, are quasi equal.

Demonstration. 40.h A : B is quasi inﬁnite. 8.h A ϩ B : B is quasi inﬁnite (componendo). 9.h A ϩ B : A is quasi equal (per conversionem rationis). Quod & c. [32, 129]

Mengoli demonstrated the other equalities in the same way, basing his proofs on Theorem 9 where, per conversionem rationis, if E : F is quasi infinite then E : E Ϫ F is quasi equal.48 Since the ratio E : F is quasi infinite, by definition 1, it can be greater than any given ratio, namely, c : c Ϫ d. By Theorem 3, ratios greater than others become per conversionem rationis smaller than others. Then per conversionem rationis, E : E Ϫ F can be smaller than the ratio c : d, which had been selected

47 ‘‘Tota magis ordinata, sibi ipsi, & alijs minu` s ordinatis, additis, vel subtractis, quasi est aequalis.’’ 48 A ratio a : b becomes per conversionem rationis a : a Ϫ b. In Theorem 41, a ratio A ϩ B : B quasi inﬁnite becomes per conversionem rationis A ϩ B:A quasi equal. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 275 to be greater than equality; that is, E : E Ϫ F can be nearer to equality than any other given ratio other than equality. Then, from Deﬁnition 3, E : E Ϫ F is a ratio quasi equality. Then, beginning with Theorem 42, Mengoli establishes ratios between all sorts of summations, quadratrices, subquadratrices, etc., and the number t. (Recall that these are all constructed using t and that these summations have t Ϫ 1 addends with different exponents.) He calculates what these ratios tend toward when the number is very large, obtaining in this way all possible quasi ratios. By way of example, consider his proof of Theorem 42:

Theorem 42. Proposition 42. Any quadratrix

aϭtϪ1 p [p ϩ 1] и as и [t Ϫ a]pϪs ͸ ͩsͪ aϭ1

is quasi equal to the tota raised to one unit larger than the one of the base where the quadratrix is found. [32, 130]49

Hypothesis. A is the quadratrix and let B tota be raised to one unit larger than the one which is the base of the quadratrix A. I say that A and B are quasi equal. [32, 130]50

Demonstration. A is equal to B, minus the addition of other totae, raised to exponents smaller than or equal to the order of the base where A is found (according to Theorem 22).

Hypoth. But B is the tota raised to an exponent one unit larger than the order of the base of A, thus the totae, raised to exponents smaller than the order of the base of A, are raised to exponents smaller than the exponent of B. 41.h As a result, A is equal to B minus the addition of other totae, raised to an order less than B. But B, minus the addition of other totae raised to 18.h orders less than B, is quasi equal to B. Consequently A is quasi equal to B. Quod & c. [32, 130]51

Notice that Mengoli bases this demonstration on Theorem 22 of the Elementum secundum and on Theorem 41. Theorem 18 is only used for the transitive property.

49 ‘‘Theor. 42. Prop. 42. Quaelibet quadratrix quasi est aequalis ad totam unitate plus ordinatam, qua`m sit eius basis.’’ 50 ‘‘Hypoth. Esto quadratrix A: & esto tota B, unitate plus ordinata, qua`m basis quadratricis A. Dico A ad B, quasi aequalem esse.’’ 51 ‘‘Demonstr. 22.2 A, est aequalis ipsi B, demptis, additisque aliqualiter acceptis totis, non plus ordinatis, hypoth. qua`m basis A. Sed B, est tota unitate plus ordinata, quam basis A: ideo` que totae, non plus ordinate qua`m basis A, sunt minu` s ordinatae qua`m B. Ergo A; est aequalis ipsi B, demptis, 41.h additisque aliqualiter acceptis totis, minu` s ordinatis, qua`m B. Sed & B, demptis, additisque 18.h aliqualiter acceptis totis, minu` s ordinatis, qua`m B, quasi est aequalis ipsi B. Ergo A, quasi est aequalis ipsi B. Quod & c.’’ 276 MA. ROSA MASSA HM 24

He raises the ﬁrst t to an order one unit bigger than the order of this base (p ϩ 1) and calls this B. A is the quadratrix in the base p. Then, by Theorem 22, A ϭ B Ϫ ͸ t n, nՅp but, by Theorem 41, B is quasi equal to B Ϫ ͸ t n, nՅp and so Theorem 18 (transitivity) yields that A is quasi equal to B. In modern notation this says that

aϭtϪ1 p [p ϩ 1] и as и [t Ϫ a]pϪs ͸ ͩsͪ aϭ1 tends to t pϩ1, when t tends to inﬁnity, because the value of the largest exponent of t is always one unit larger than the order of the base where it is found. Considering only the quasi ratios of the quadratrices of the lateral of the quadratrix table (Fig. 4), that is, O.ap, Theorem 42 is reminiscent of the equation

aϭt ͸ ap aϭ1 1 lim ϭ t pϩ1 p ϩ 1 when t tends to inﬁnity. Roberval [43, 171; 5, 19] and Wallis [38, 34; 44, 384] announced similar results for the cases p ϭ 1, 2, and 3, from which they inferred the validity of the general law.52 The differences between the work of Roberval and Wallis on the one hand and Mengoli on the other are: (1) Roberval and Wallis

52 In this connection, Roberval [43, 171–173; 5, 19–20], in his Traite´ des indivisibles (1634, published in 1693), argued that

1 ϩ 2 ϩ 3 ϩ ....ϩtϭsAt2 ϩsAt and that the last term could be ignored when t is very large. He argued similarly relative to the sum of the squares

12 ϩ 22 ϩ .....ϩt2 ϭdAt3 ϩsAt2 ϩhAt, contending that the two last terms could be ignored when t is very large. Roberval gave no kind of justiﬁcation or mathematical proof of these statements, but just veriﬁed that they were true by giving values and presenting geometric examples such as triangles or cubes. Likewise, in his discussion of the ratio 02 ϩ 12 ϩ 32 ϩ ...... ϩn2 , n2 ϩn2 ϩn2 ϩ...... ϩn2

Wallis concluded ‘‘that the required ratio was rather more than dA, and that the excess over dA diminished as the number of terms constituting the series increased’’ ([38, 32; 44, 382]). Both Roberval and Wallis used these results to find areas. Mengoli, in contradistinction, did not verify his summations by giving values or by presenting geometric examples. HM 24 MENGOLI ON ‘‘QUASI PROPORTIONS’’ 277 performed calculations only for p small, whereas Mengoli considers p any natural number; and (2) Roberval and Wallis add t terms of a sum, while Mengoli sums up t Ϫ 1 terms. Relative to this second point, however, in the limit the results are the same because the powers of smaller degrees are ignored. Cavalieri, in Proposition XXIII of the Exercitatione quarta [17, 279], also proved this general result geometrically, and Mengoli clearly knew this work by his master very well.53 As noted above, Pascal and Fermat deduced this result directly from the sum of the powers and mentioned its use for quadratures. Mengoli, in contrast, sets up the theory of quasi proportions to justify the result, which, of course, he already knew. When, in Theorem 42, Mengoli calculates that the quadratrix and the number t raised to one unit larger than the order of the base of the quadratrix were quasi equals, he might naturally have used this quasi equality directly in order to calculate the area of the curve associated to the quadratrix. In fact, he does not do this. He does not try to find the area of this figure directly through the value of the summation when the number of lines or rectangles increases, and in this way avoids the problems that Cavalieri faced in having to justify whether ‘‘all the lines’’ were equal to the figure. Mengoli, unlike Cavalieri, never compares two figures through the compari- son of lines, nor does he superimpose figures; rather, he establishes quasi ratios between figures.

CONCLUDING REMARKS Mengoli’s originality lies not only in the way in which he presented this work, but also in the content of the work. His highly innovative numerical theory, which incorporates the new idea of quasi ratio, made it possible to calculate limits and then to do quadratures. He devised it at a time when geometric methods still dominated and few mathematicians introduced algebraic elements to do geometry. By using letters to symbolize numbers and by using triangular tables as Vie`te had done, he was able to generalize some results. Mengoli did not calculate the summation of powers in terms of definite values. Instead, he invented an original and useful construction involving summations. He then placed these summations in triangular tables and obtained new relations between the terms of these tables. In this way, he could calculate the summations of positive integer powers and summations of products of powers indefinitely. From these calculations, Mengoli used the original idea of quasi ratio to study the value of these summations when the number of terms of the sum increases. And once more he did not do this by giving values. Using the properties of quasi proportions, the calculations of summations, and the various tables, he could calculate countless quasi ratios. His originality in applying this theory to geometry lay in finding countless areas of figures associated with algebraic expressions. The table of quadratures, which he constructed in Elementum sextum, could be extended as far as desired; that is, Mengoli did not find areas of

53 On Cavalieri, see [11, 450; 40, 217]. 278 MA. ROSA MASSA HM 24 concrete rectangles or volumes of pyramids. Wishing to generalize, he calculated areas of plane ﬁgures in the interval (0, 1), expressible as

p (p ϩ 1) и и xs и (1 Ϫ x) pϪs. ͩsͪ

For Mengoli, then, the purpose both of the theory of quasi proportions and of the triangular tables was, above all, to serve as a tool for obtaining countless limits and countless quadratures. In reference to Roberval, Evelyn Walker argued that the Frenchman ‘‘deserves full credit for having been one of the ﬁrst to use the sum of the power series as a basis for reckoning with inﬁnitesimals’’ [43, 44]; the same could be said of Mengoli. In spite of all these innovations, Mengoli was scarcely understood. In a letter to Collins, Isaac Barrow said that Mengoli’s style was harder than Arabic, and that if Mengoli had found something new, he did not have the time to investigate it [22, 49]. Even though his reputation was strong during his lifetime, it seems that Mengoli died isolated and ignored. The reasons for this are not clear. It is possible that his complex and confusing writing style and the complicated nature of his notation made his works too hard to read; perhaps, for this reason, he had no followers. It is equally possible that his introduction of algebra into geometry failed to accord with the prevailing mathematical practice of the 17th century.

ACKNOWLEDGMENTS I thank the Bodleian Library for providing microﬁlms of the books of Mengoli. I am also grateful to Albert Dou, Eberhard Knobloch, and Catherine Goldstein, each of whom read earlier versions of this article and made many remarks concerning content and language. Teresa Mejo´ n, Toni Malet, and Albert Dou offered invaluable assistance with the translation of the Geometriae, and Laura Martorell helped with the English language. Special thanks, however, go to Toni Malet for having initially suggested that I work on Pietro Mengoli, for reading earlier versions of this article, and for proposing many helpful suggestions concerning content and language.

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