Euler takes the concept of Latin square (an

Euler gives hundreds of examples of Latin and Greco-Latin squares and takes many lengthy detours through this paper, asking questions about Latin squares in which the diagonals also satisfy the "Latin square" property. In the end, he argues, but fails to prove rigorously, that no Greco-Latin square of size 4

According to the records, it was read to the St. Petersburg Academy on March 8, 1779.

- Originally published in
*Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen*9, Middelburg 1782, pp. 85--239 - Also available in
*Commentationes arithmeticae*2, 1849, pp. 302--361 *Opera Omnia*: Series 1, Volume 7, pp. 291--392

- Original publication (this is the best copy available at this time. A few pages are difficult to read, and pp. 150, 151, 236, and 237 are missing):
- A higher-quality version of the original article is available via Archive.org: E530.
- A Translation of E530 by Andie Ho and Dominic Klyve is available.
- This is the only one of Euler's papers to be published in a Dutch journal.

- E530 is discussed in Ed Sandifer's How Euler Did It April 2006 column published online by the MAA.
- An analysis of E530, along with a survey on Greco-Latin squares by Klyve and Stemkoski appears in the January 2006 volume of the College Mathematics Journal.
- This is one of Euler's more popular papers, having been cited 35 times in the last 60 years. 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 E530 include:
- Afsarinejad KON., "The optimality of knut vik designs." Utilitas Mathematica, 41, pp. 91-96 (May 1992).
- Alter R., "How many Latin squares are there." Am. Math. Mon., 82 (6), pp. 632-634 (1975).
- Argunov BI, Emelchenkov EP., "Incidencestructures and ternary-algebras." Russ. Math. Surv., + 37 (2), pp.1-39 (1982).
- Atkin AOL, Hay L, Larson RG., "Construction of knut vik designs." J. Stat. Plan. Infer., 1 (3), pp. 289-297 (1977).
- Atkin AOL, Hay L, Larson RGE., "Numeration and construction of Pandiagonal Latin squares of prime-order." Comput. Math. Appl., 9 (2), pp. 267-292 (1983).
- Bose RC, Shrikhande SS., "On the falsity of Euler's conjecture about the non-existence of 2 orthogonal Latin squares of order 4T+2." Starp. Natl. Acad. Sci. USA, 45 (5), pp. 734-737 (1959).
- Colbourn CJ, Dinitz JH., “Mutually orthogonal latin squares: a brief survey of constructions.” Journal of Statistical Planning and Inference, 95 (1-2), pp. 9-48 (Sp. Iss. May 1 2001).
- David HA., "First (questionable) occurrence of common terms in mathematical-statistics." Am. Stat., 49 (2) pp. 121-133 (May 1995).
- Deza M, Vanstone SA., "Bounds for permutation arrays." J. Stat. Plan. Infer., 2 (2), pp. 197-209 (1978).
- Finney DJ., "Latin squares of the 6th order." Experientia, 2 (10), pp. 404-405 (1946).
- Finney DJ., "Some orthogonal properties of the 4X4 and 6X6 Latin squares." Ann. Eugenic, 12 (4), pp. 213-219 (1945).
- Ghosh SP., "Quadrics in finite Euclidean geometries and application to construction of balanced incomplete block designs." SIAM J. Appl. Math., 19 (4), pp. 730-& (1970).
- Gipser T, Jager HA, Rapp L., "Broadcasting, scalability, and reconfigurability aspects in an all-optical network architecture." Fiber Integrated Opt., 17 (1), pp. 21-40 (1998).
- Hanani H., “The existence and construction of balanced incomplete block-designs.” Annals of Mathematical Statistics, 32 (2), pp. 361-386 (1961).
- Hedayat A., "Complete solution to existence and nonexistence of knut-vik designs and orthogonal knut-vik designs." J. Comb. Theory A, 22 (3), pp. 331-337 (1977).
- Hedayat A., "On a statistical optimality of magic squares." Stat. Probabil. Lett., 5 (3), pp. 191-192 (Apr 1987).
- Hedayat A., "Set of 3 mutually orthogonal Latin squares or order-15." Technometrics, 13 (3), pp. 696-& (1971).
- Johnson DM, Dulmage AL, Mendelsohn NS., "Orthomorphisms of groups and orthogonal Latin squares .1." Canadian J. Math., 13 (3), pp. 356-& (1961).
- Kirton HC., "Mutually orthogonal partitions of the 6X6 Latin squares." Utilitas Mathematica, 27 (MAY), pp. 265-274 (1985).
- Knuth DE., "Mathematics and computer science - coping with finiteness." Science, 194 (4271), pp. 1235-1242 (1976).
- Lam CWH., "The search for a finite projective plane of order 10." Am. Math. Mon., 98 (4), pp. 306-318 (Apr 1991).
- Mullen GL., "A candidate for the next Fermat problem." Math. Intell., 17 (3) pp. 18-22 (Sum. 1995).
- Parker ET., "Maximum number of digraph-distinct ordered quadruples on 6 marks." J. Comb. Theroy A, 19 (2), pp. 245-246 (1975).
- Parker ET., "Orthogonal Latin squares." P. Natl. Acad. Sci. USA, 45 (6), pp. 859-862 (1959).
- Pullman NJ, Shank H, Wallis WD., Clique coverings of graphs .5. maximal-clique partitions B." Aust. Math. Soc., 25 (3), pp. 337-356 (1982).
- Rivest RL., "Permutation polynomials modulo 2(W)." Finite Fileds Th. App. 7 (2), pp.287-292 (Apr 2001).
- Roberts CE., “Sets of mutually orthogonal latin squares with like subsquares.” Journal of Combinatorial Theory Series A, 61 (1), pp. 50-63 (Sep 1992).
- Stinson DR., The existence of Howell designs of odd side." J. Comb. Theory A, 32 (1) pp. 53-65 (1982).
- Uko LU., “The anatomy of magic squares.” ARS Combinatoria, 67, pp. 115-128 (Apr 2003).
- Ullrich P., “Officers, playing cards, and sheep - on the history of Eulerian squares and of the design of experiments.” Metrika, 56 (3), pp. 189-204 (2002).
- Webb S., "Interactions between experiment designer and computer." Nav. Res. Logist. Q., 16 (3), pp. 423-& (1969).
- Wijshoff HAG, Vanleeuwen JON., "Linear skewing schemes and
*d*-ordered vectors." IEEE T. Comput., 36 (2), pp. 233-239 (Feb. 1987). - Wojtas M., "Six mutually orthogonal Latin squares of order 48." J. Comb. Des., 5 (6) pp. 463-466 (1997).
- Yin GZ, Jillie DW., "Orthogonal design for process optimization and its application in plasma-etching." Solid State Technol., 30 (5), pp. 127-132 (May 1987).