E407  Problema algebraicum ob affectiones prorsus singulares memorabile
(An algebraic problem that is notable for some quite extraordinary relations.)
Summary:
Eneström says that this work contains orthogonality conditions for rectangular corrdinates in space. He also says that Euler will express the coefficients of linear homogeneous
transformations that let a sum of n square numbers remain constant using trigonometric
functions for n = 3, 4, and 5, and also using rational parametric functions for n = 3 and 4.
According to the records, it was presented to the St. Petersburg Academy on March 5, 1770.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 15, 1771, pp. 75106

Opera Omnia: Series 1, Volume 6, pp. 287  315
 Reprinted in Commentat. arithm. 1, 1849, pp. 427443 [E407a]
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
Documents Available:
 Original Publication: E407
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E407 include:
 Dorier JL., “A general outline of the genesis of vectorspace theory.” Historia Mathematica, 22 (3), pp. 227261 (Aug 1995).
 Shuster MD., “A survey of attitude representations.” Journal of the Astronautical Sciences, 41 (4), pp. 439517 (OctDec 1993).
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