**From:**

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**Date:**

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**Summary:**

Sends hints. Will give explanation at Wednesday's meeting of 'Committee on Charles Babbage.'

**Contributor:**

Henry Warburton

Sir John Herschel

n.d.

RS:HS 18.88

Sends hints. Will give explanation at Wednesday's meeting of 'Committee on Charles Babbage.'

John Herschel Project

Henry Warburton

Sir John Herschel

[24 April 1816]

RS:HS 18.36

Suggests changes for JH's paper on Swedish felspar submitted to Geological Society. Confusion between this and other silicates. Compares JH's analysis to that of Wilhelm Hisinger and others.

John Herschel Project

Henry Warburton

Sir John Herschel

[3 June 1816]

RS:HS 18.37

Discussed JH's proposed felspar paper [see HW's 1816-4-24] with W. H. Wollaston. Suggests using Wollaston's and [recently deceased Smithson] Tennant's unpublished method to test for alkalis and give Tennant credit for it. James Smithson first coined term 'silicate'.

John Herschel Project

Henry Warburton

Sir John Herschel

[6 February 1829]

RS:HS 18.38

Apologizes for delay in writing out what JH requested. He will have it tomorrow.

John Herschel Project

Henry Warburton

Sir John Herschel

[19 October 1847]

RS:HS 18.39

Sends copy of HW's paper on combinations of 'Plural Elements,' printed in Transactions of Cambridge Philosophical Society, based on the method of continued subtraction suggested by Augustus De Morgan. Shows solution for Bernoulli's fifth number.

John Herschel Project

Henry Warburton

Sir John Herschel

[14 November 1847]

RS:HS 18.40

Close to finding expeditious, symmetric method for computing Bernoulli's numbers. Resolves four formulas that HW sent to JH earlier today.

John Herschel Project

Henry Warburton

Sir John Herschel

[22 November 1847]

RS:HS 18.41

Thanks for comments on HW's paper and work on properties of Bernoulli's numbers. Claims HW's method of continued subtraction is convenient process for determining numerical coefficients. Found error, long perpetuated in literature, regarding Bernoulli's 13th number. Offers correct solution.

John Herschel Project

Henry Warburton

Sir John Herschel

[23 November 1847]

RS:HS 18.42

More calculations showing continued subtraction method in solution of Bernoulli's 13th number.

John Herschel Project

Henry Warburton

Sir John Herschel

[24 November 1847]

RS:HS 18.43

More calculations using continual subtraction method to test Bernoulli's 15th number.

John Herschel Project

Henry Warburton

Sir John Herschel

[25 November 1847]

RS:HS 18.44

More continual subtractions to resolve Bernoulli's 17th number. Error in official value given in Encyclopaedia Metropolitana. Also working on Bernoulli's 18th number.

John Herschel Project

Henry Warburton

Sir John Herschel

[29 November 1847]

RS:HS 18.45

Will submit paper to Cambridge [Philosophical] Society revising all [Leonhard] Euler's values for Bernoulli's numbers. Doubts accuracy of values up to B=49 quoted by George Peacock.

John Herschel Project

Henry Warburton

Sir John Herschel

[5 December 1847]

RS:HS 18.46

Error in Leonhard Euler's value for Bernoulli's 21st number. Promotes HW's method of continued subtraction as a reliable means of proving accuracy of values for higher Bernoulli numbers.

John Herschel Project

Henry Warburton

Sir John Herschel

[13 December 1847]

RS:HS 18.47

Destroy all HW's letters claiming errors in Leonhard Euler's values for Bernoulli's numbers. Mr. Hensley of Trinity College found weakness in HW's continued subtraction method. Clarifies symbols HW used to represent P. S. Laplace's expressions of Bernoulli's numbers.

John Herschel Project

Henry Warburton

Sir John Herschel

[14 December 1847]

RS:HS 18.48

Continues explicating P. S. Laplace's formula for determining Bernoulli's numbers. Concerned that JH sees HW claiming too much for HW's continued subtraction method.

John Herschel Project

Henry Warburton

Sir John Herschel

[20 December 1847]

RS:HS 18.49

Verified Leonhard Euler's values for Bernoulli's numbers up to B=25 by method previously outlined and to be stated fully in this letter. Mr. Hensley [of Trinity College] proposes to revise incorrect values given in Penny Cyclopaedia. Computing a Bernoulli number by P. S. Laplace's formula takes three days. HW's method of continued subtraction requires only 24 hours.

John Herschel Project

Henry Warburton

Sir John Herschel

[23 December 1847]

RS:HS 18.50

Proved accuracy of Leonhard Euler's values for Bernoulli's 27th number. Plans to test number 29. Will investigate Thomas Clausen's theorem, the generality of which JH appears to have disproved.

John Herschel Project

Henry Warburton

Sir John Herschel

[30 December 1847]

RS:HS 18.51

Demonstrates property of 'Coefficients of the terms of the Numerators of the Fractions which generate the odd powers of the Numbers of the Natural Series.' Claims this method can compute Bernoulli's numbers 37, 39, and 41.

John Herschel Project

Henry Warburton

Sir John Herschel

[28 January 1850]

RS:HS 18.52

Thanks for citing (18 months ago) John Brinkley's paper on 'General Term.' Heard that Augustus De Morgan notified JH of deductions HW drew from Brinkley's theorems, extending them into permutations and combinations. Re-read JH's ['On the Development of Exponential Functions' (1816)]. Asks where to buy JH's Examples in Finite Differences [1820] for HW's great-nephew [Howard Elphinstone].

John Herschel Project

Henry Warburton

Sir John Herschel

[31 January 1850]

RS:HS 18.53

Thanks for sending Examples in Finite Differences [1820] to WH's great-nephew Howard Elphinstone. John Brinkley's paper opened up new possibilities for permutations and combinations.

John Herschel Project

Henry Warburton

Sir John Herschel

[24 February 1850]

RS:HS 18.54

Admires chimes analogy in JH's 'On Circulating Functions' [1818]. Resumed interest in partitions of numbers in development of periodic functions. Finds JH's method superior. Encourages him to publish it. WH and Augustus De Morgan developed original method, which WH erred in attributing to Leonhard Euler. De Morgan's paper to Cambridge [Philosophical] Society demonstrates theorem leading to proof of Euler's theorem.

John Herschel Project