The repercussion of José Anastácio da Cunha in Britain and USA in the nineteenth century

Abstract

José Anastácio da Cunha (1744–87) is usually recognized as one of the most important Portuguese mathematicians ever. Nevertheless, his work does not seem to have had much repercussion abroad. His Principios mathematicos (Lisbon, 1790) were translated into French, but with very limited impact. However, thanks to a review by John Playfair, an English textbook by John Radford Young featured a proof by Anastácio da Cunha (of a proposition on parallels) and a definition of proportion influenced by the one used by Cunha. This proof on parallels also made its way to an American textbook by Benjamin Greenleaf. Young’s and Greenleaf’s are, so far, the only known cases of actual use (instead of mere reference to) of Cunha’s work outside Portugal.

To Maria Fernanda Estrada, with friendship.

This research was financed by FEDER Funds through Programa Operacional Factores de Competitividade – COMPETE and by Portuguese Funds through FCT – Fundação para a Ciência e a Tecnologia, within the Project Est–C/MAT/UI0013/2011.

A partial version of this paper has been published in the Proceedings of a Research Seminar on History and Epistemology of Mathematics held in Braga, Portugal, on 20 May 2010, for the 80th birthday of Maria Fernanda Estrada.

Notes

¹Cunha’s Ensayo sobre os principios de mechanica (Essay on the principles of mechanics) has been translated into French recently (Cunha 2004); and a couple of manuscripts in French and English have been published in Ralha et al. (2006).

²This reissue does not mean that the book was successful: on the contrary, it appears to be a false reprint, making use of the remainders of the 1811 edition with new cover and title pages.

³‘rigorous analytic definition of differential’

⁴The first three of these reviews are collected in Ferraz et al. (1990) and the fourth is reprinted in Domingues (2011).

⁵In Duarte and Silva (1987) there are several examples of its influence on nineteenth-century Portuguese mathematicians.

⁶In the first edition, Playfair had written ‘two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another’ (Playfair 1795, 1st ed, 7). Both of these statements are immediately equivalent to the nowadays more usual ‘through a given point at most one parallel can be drawn to a given straight line’—but they are not equivalent to the also usual ‘through a given point one and only one parallel can be drawn to a given straight line’.

⁷Playfair was probably more influential in geology than in mathematics, thanks to his Illustrations of the Huttonian theory of the Earth, which did much to clarify James Hutton’s geological theory. Playfair’s entry in the Dictionary of scientific biography is almost entirely dedicated to his work as a geologist (Challinor 1975).

⁸Synthesis and (especially) analysis are terrible words with variable meanings. Ackerberg-Hastings (2002) lists three senses for the distinction, used in Playfair’s time: (1) mathematical styles, synthesis referring to a more classical, geometrical, style, and analysis referring to a more modern style, making use of algebraic symbolism; (2) the original sense, referring to methods of proof (or, more usually, synthesis as a method of proof and analysis as a method of investigation); and (3) educational techniques, synthetical proofs being appropriate for training logical reasoning and analytical investigations being simpler, more applicable, and more inducive to the spirit of discovery.

⁹We have seen in the section ‘British influence on Anastácio da Cunha and an early publication about him in Britain’ that there was considerable British influence on Cunha. He may have been a little too ‘British’ for Playfair.

¹⁰Incidentally, this proof is a simple but impeccable - argument.

¹¹I thank Tony Crilly for sharing these census data with me.

¹²He even found ‘somewhat remarkable, that late writers on geometry have not availed themselves of this decided improvement’.

¹³Foundationally, Young (1831a) rests on a naïve consideration of limits supported by general power-series expansions, clearly influenced by French authors such as Lagrange and Lacroix.

¹⁴Following Heath’s version, which in this respect is close to Simson’s.

¹⁵‘Continentur in prop. 32. Neque ante illam adhibentur.’

¹⁶Playfair mentions this rule to apologize for departing from it in his own supplement on solid geometry, ‘assuming the existence of such solids, or such lines as are evidently possible’ (1795, x), in order to shorten and simplify it; but the point here is that he did acknowledge the rule, so that presumably it would be preferable to follow it, unless not doing so led to significant simplification (and, of course, the objects considered were ‘evidently possible’).

¹⁷Heath’s version is somewhat different, especially in the case of definition 5: admittedly closer to being literal, and therefore shorter, but also a little less clear. Our concern here is not how, c. 300 bce, Euclid conceived of and handled ratios and proportions (Grattan-Guinness 1996); rather, we are interested in eighteenth- and nineteenth-century versions of such classical mathematics.

¹⁸In modern notation, has the same ratio to which has to (or, using definition 6, are proportionals) if, taking any equimultiples of and , and any equimultiples of and ,

¹⁹Other translators would write that those are proportional magnitudes, or that those magnitudes are in proportion.

²⁰This aspect of Barrow’s argument is weak: Heath (1926, II, 292) explains this absence remarking that such a definition would not be more than a particular case of V. def 3; also, propositions 14 and 17 of book VII do speak of ratios, clearly applying the general definition to numbers.

²¹‘He justo, hé necessario que nas definições não haja escuridades’ (Ralha et al. 2006, II, 18–19); this is Cunha’s justification in the manuscript foreword mentioned above.

²²This is Playfair’s translation (1812, 428), except that Playfair, certainly by mistake, finishes with ‘these numbers are called proportionals’. Otherwise it is a fair translation (neglecting the detail that Cunha meant ‘proportional’ to be an adjective, rather than a noun). Cunha’s original reads

Se humas antecedentes, e suas consequentes forem taes, que em nenhuma antecedente possa caber submultiplice algum da sua consequente mais vezes do que em qualquer outra antecedente cabe hum igualmente submultiplice da sua consequente; chamar-se haõ essas antecedentes, e consequentes, proporcionaes (Cunha 1790, 20).

²³Denoting integer division by , we can say that Cunha calls proportional if, taking any equisubmultiples of and , that is, while Playfair calls proportionals if, taking any equimultiples of and , that is,

²⁴Duae rationes (A ad B, et C ad F) sunt similes, aequales, eaedem; cum unius antecedens (A) aeque seu eodem modo (hoc est nec magis, nec minus) continet suum consequens (B), quo alterius antecedens (C) continet suum consequens (F).

Vel quando unius antecedens (A) eodem modo continetur in suo consequente (B), quo (C) antecedens alterius in suo (D). (Tacquet 1665, 132)

²⁵Rationes aequales sunt […] quando et consequentes ipsae, et consequentium similes partes aliquotae quaecunque, in antecedentibus aequali semper numero continentur. (Tacquet 1665, 136)

²⁶In an interpretation for modern readers, with standing for integer division, we may say that Cunha, assuming that are not proportional (see footnote 23 on page 13), first takes submultiples and such that (the case being analogous); if , but ; then he takes multiples and such that and ; it is then possible to consider a multiple such that ; now, from follows and, since , ; thus, but , which means that the conditions of Euclid’s V. def 5 do not apply.

Young, on the other hand, also assuming that are not proportional, simply takes multiples and such that ; if , then but .

This modern rendering suggests that it would have been much easier for Cunha to conclude, from but , that but . But, instead of ‘ but ’, Cunha had ‘ but ’, where ‘H is the largest multiple of contained in ’, and so on. He should have followed the simpler path if he were working things out in algebraic notation (which he mastered pretty well) and then translating back to Euclidean-style language; but clearly he was not. This is yet another example of how algebraic-language renderings of Euclidean-style arguments can be historically deceptive (Grattan-Guinness 1996).