A forgotten booklet by Goldbach now rediscovered and three versions of its contents

Abstract

Dedicated to the memory of Jacqueline Stedall

Acknowledgements

I am deeply grateful to my colleague Olle Axling at Linköping University, who after my tip looked for the booklet at Linköpings Stiftsbibliotek and found it!

I will also send my deep gratitudes to Siegmund Probst for wonderful discussions on the subject of the article and for his great help with translation.

Disclosure statement

No potential conflict of interest was reported by the author.

Reproduction of the seven pages of the original paper. Reproduced with permission of Linköpings Stiftsbibliotek.

 

Notes

¹ The Latin text is: ‘Legi nuper sine magna animi voluptate recensitam in Actis Eruditorum Algebram tuam iis ipsis verbis quibus id fieri optaveram; adjungerunt deinde specimen methodi ad summas serierum, quod nosti, in a quo insertum sit his actis haud equidem seio, sed dilatum mallem donec methodus paulo curatius exculta provectaque esset.’

² Eneström (1887) supposed that Goldbach’s use of the symbol ∞ as a symbol for infinity was very early. Modern research tells that Wallis defined the symbol already in 1655 in his De sectionibus conicis.

³ The area is , and we can easily recognize its origin from the series for the circle with diameter 1 that Leibniz introduced in the article, namely : The terms can be subtracted pairwise and then divided by two.

⁴ In the quoted letter to Daniel Bernoulli (Fuss 1843, 186–187) Goldbach writes that he was aware of the equality and its relation to the Leibniz series.

⁵ The Latin text is: ‘Quod nuper Holmiæ evulgatum est Specimen Methodi ad Summas Serierum, clarissimus et humanissimus Auctor C.G. ad me transmisit, cum aliis rebus in Germania differrer’.

⁶ Cf. Simon Foucher to Leibniz, 31 December 1691 (Leibniz 2009, 474): ‘Mr Osannan dit qu’il est vray que vous luy avez donné l’ouverture de sa quadrature du cercle, mais il se pleint de ce qu’ayant esté trop lent à decouvrir ce que vous en sçaviez vous luy avez donné lieu de faire là dessus ses meditations et d’en trouver ce qu’il en a trouvé. De sorte qu’il pretend avoir droit aussi bien que vous à cette decouverte’.