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XII. Note on the History of certain Formula in Spherical Trigo- 

 nometry. By I. Todhunter, M.A., F.R.S, 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 



THERE are four formulas in Spherical Trigonometry which 

 are usually called Gauss's Theorems or Gauss's Analogies. 

 These formulas are 



sin J «cos^(B — C) = sin -JA sin \{b + c) 



and three others of a like nature. 



The formulas, however, are really due to Delambre ; but in 

 consequence of an erroneous reference, his claim has been ob- 

 scured, and mathematicians have been put to inconvenience in 

 investigating the matter. 



Gauss printed the formula? in 1809 in his Theoria Motus, 

 p. 51. He says they would be sought for in vain in books on 

 trigonometry; he omits the demonstration for the sake of 

 brevity. 



Delambre, in the first volume of his Astronomie, published in 

 1814, claims the formulas (see his pages 164 and 195); in both 

 places he refers to the Connaissance des Terns for 1808. The re- 

 ference, however, should be to the Connaissance des Terns for 

 1809, which was published in April 1807. Here the four for- 

 mulas are given without demonstration, together with some 

 others which follow immediately from them. One of these 

 other formulas is 



t , n i„- faMi(6- e )Mni(B + C) . 

 tan * a rini(B-C) ' 



this Delambre ascribes to M. Henri. 



As Delambre is here discussing the solution of an astrono- 

 mical problem given by M. Henri, it might at first sight have 

 appeared probable that after this formula had been used by M. 

 Henri, the four formulas improperly ascribed to Gauss were in- 

 vestigated by Delambre. But it may be inferred from the top 

 of page 446 of Delambre' s remarks, that he had been previously 

 acquainted with the formulas. 



Delambre claims the formulas and gives the correct reference 

 on p. 349 of the Connaissance des Terns for 1812, which was 

 published in July 1810 : Delambre is here reviewing Gauss's 

 Theoria Motus. 



Delambre's erroneous reference has been adopted by some 

 writers. Thus Bowditch says od p. 737 of the first volume of 

 his translation of the Mecanique Celeste, " Delambre, in his As- 

 tronomie, vol. i. p. 164, observes that he had given several of 



