History of calculus
Leibniz explains his differential method of tangents in a letter from Hanover
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Original prompt & settings (JSON) ↗Newton hid his core method inside an anagram; Leibniz answered by writing out his own differential notation and an example.
Hanover, 21 June 1677
Leibniz dated a letter at Hanover, 21 June 1677, explaining his method and inverse method of tangents — what would now be called differential and integral calculus.
He used dx and dy for small corresponding changes in coordinates and related them to the direction of a curve's tangent. He showed the method could handle equations containing irrational expressions, and supplied examples of inverse tangent problems.
Why he wrote it
Leibniz was replying after finally receiving Newton's letter of 24 October 1676. Newton had asserted a broad method for tangents, quadratures, and inverse problems while withholding the algorithm itself, concealing key details in an anagram.
Leibniz answered by describing his own notation and procedure openly.
A letter, not a publication
Robert Hooke copied the communication. That five-page manuscript, with mathematical formulae and diagrams, survives in the Royal Society archive as EL/L5/99.
The algorithm entered the London correspondence record but was not a public journal publication; Leibniz's first printed account of differential calculus followed in 1684.
'Complete solution to the tangent problem' is a later characterization. The document gives a general differential method and some inverse examples, not a proof that every direct and inverse tangent problem had been solved.
Sources
Researched 24 Aug 2026 5 sources not yet audited Date corrected
How this was checked
Researched from the web into a fact sheet and rewritten from that sheet. This article has not yet had an independent model audit. How the pipeline works →
Date corrected. The research places this at 1677-06-21.
What the sources leave uncertain
- The date and mathematical content are well documented. The phrase “complete solution to the tangent problem,” however, is a later characterization rather than the title of a 1677 publication.
- The Royal Society catalogue labels the surviving item a copy of a letter to Collins, while histories of the exchange describe it as Leibniz’s reply sent through Oldenburg. This affects the description of the correspondence chain, not the date or mathematical content.