Geometry and infinite zeros
Grandi
Guido Grandi (1671–1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into produced varying results: either
or
Grandi's explanation of this phenomenon became well known for its religious overtones:
In fact, the series was not an idle subject for Grandi, and he didn't think it summed to either 0 or 1. Rather, like many mathematicians to follow, he thought the true value of the series was 1⁄2 for a variety of reasons.
Grandi's mathematical treatment of occurs in his 1703 book Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita. Broadly interpreting Grandi's work, he derived through geometric reasoning connected with his investigation of the witch of Agnesi. Eighteenth-century mathematicians immediately translated and summarized his argument in analytical terms: for a generating circle with diameter a, the equation of the witch y = a3/(a2 + x2) has the series expansion
and setting a = x = 1, one has 1 − 1 + 1 − 1 + · · · = 1⁄2. According to Morris Kline, Grandi started with the binomial expansion
and substituted x = 1 to get . Grandi "also argued that since the sum was both 0 and 1⁄2, he had proved that the world could be created out of nothing."
Grandi offered a new explanation that in 1710, both in the second edition of the Quadratura circula and in a new work, De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica. Two brothers inherit a priceless gem from their father, whose will forbids them to sell it, so they agree that it will reside in each other's museums on alternating years. If this agreement lasts for all eternity between the brother's descendants, then the two families will each have half possession of the gem, even though it changes hands infinitely often. This argument was later criticized by Leibniz.
The parable of the gem is the first of two additions to the discussion of the corollary that Grandi added to the second edition. The second repeats the link between the series and the creation of the universe by God:
Marchetti
After Grandi published the second edition of the Quadratura, his fellow countryman Alessandro Marchetti became one of his first critics. One historian charges that Marchetti was motivated more by jealousy than any other reason. Marchetti found the claim that an infinite number of zeros could add up to a finite quantity absurd, and he inferred from Grandi's treatment the danger posed by theological reasoning. The two mathematicians began attacking each other in a series of open letters; their debate was ended only by Marchetti's death in 1714.
Leibniz
With the help and encouragement of Antonio Magliabechi, Grandi sent a copy of the 1703 Quadratura to Leibniz, along with a letter expressing compliments and admiration for the master's work. Leibniz received and read this first edition in 1705, and he called it an unoriginal and less-advanced "attempt" at his calculus. Grandi's treatment of 1 − 1 + 1 − 1 + · · · would not catch Leibniz's attention until 1711, near the end of his life, when Christian Wolff sent him a letter on Marchetti's behalf describing the problem and asking for Leibniz's opinion.
Background
As early as 1674, in a minor, lesser-known writing De Triangulo Harmonico on the harmonic triangle, Leibniz mentioned very briefly in an example:
Presumably he arrived at this series by repeated substitution:
And so on.
The series also appears indirectly in a discussion with Tschirnhaus in 1676.
Leibniz had already considered the divergent alternating series as early as 1673. In that case he argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite. Two years after that, Leibniz formulated the first convergence test in the history of mathematics, the alternating series test, in which he implicitly applied the modern definition of convergence.
Solutions
In the 1710s, Leibniz described Grandi's series in his correspondence with several other mathematicians. The letter with the most lasting impact was his first reply to Wolff, which he published in the Acta Eruditorum. In this letter, Leibniz attacked the problem from several angles.
In general, Leibniz believed that the algorithms of calculus were a form of "blind reasoning" that ultimately had to be founded upon geometrical interpretations. Therefore, he agreed with Grandi that claiming that the relation was well-founded because there existed a geometric demonstration.