## Early Papers on Interval Computations

### Origins of Interval Computations: from Archimedes to 1960s

Achimedes used two-sided bounds to compute Pi: Archimedes, "On the measurement of the circle", In:
Thomas L. Heath (ed.), *The Works of Archimedes*, Cambridge
University Press, Cambridge, 1897; Dover edition, 1953, pp. 91-98.

The concept of a function having values which are bounded
within limits was discussed by W. H. Young: W. H. Young, "Sull
due funzioni a piu valori constituite dai limiti d'una funzione di
variable reale a destra ed a sinistra di ciascun punto",
*Rendiconti Academia di Lincei, Classes di Scienza Fiziche*,
1908, Vol. 17, No. 5, pp. 582-587.

For positive quantities,
rules for the arithmetic of intervals were explicitly stated and
applied to evaluation of rational expressions by Vladimir M.
Bradis^{*}: V. M. Bradis, "An experience of the
verification of practically useful operations with approximate
numbers", Proceedings of Tver Pedagogical Institute, 1927, No. 3,
in Russian.

The concept of operations with a set of
multi-valued numbers was introduced by R. C. Young, who developed a
formal algebra of multi-valued numbers: A
book on interval calculus appeared in Yugoslavia in 1932:
The special case of closed intervals (not necessarily positive) was
further developed by P. S. Dwyer: Interval mathematics was further developed by M. Warmus:
by T.
Sunaga:
T. Sunaga, "Theory of interval algebra and
its application to numerical analysis", In: *Research
Association of Applied Geometry (RAAG) Memoirs*, Ggujutsu Bunken
Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29-46 (547-564);
reprinted in Japan Journal on Industrial and Applied Mathematics,
2009, Vol. 26, No. 2-3, pp. 126-143.

and by R.
E. Moore
For an early history of interval computations, see also

S. Markov and K. Okumura, "The Contribution of T. Sunaga to
Interval Analysis and Reliable Computing", In: T. Csendes (ed.),
*Developements in Reliable Computing*, Kluwer, Dordrecht,
1999, pp. 167-188.

### Selected Papers from the 1970s and 1980s

^{*}This approach was further developed in several other publications of
V. M. Bradis:

- V. M. Bradis, "Theory and Practice of Computations", Moscow, Uchpedgiz Publ.,
1937
(in Russian).
- V. M. Bradis, "Tools and Methods of Elementary Computations", Moscow, Russian
Academy of Pedagogical Sciences, 1948 (in Russian).
- V. M. Bradis, Oral and Written
Computations: Tools and Techniques for Computations,
in: "Encyclopedia of Elementary Mathematics", Moscow, 1951
(in Russian), see Sections 6 and 8; German translation:
H. Grell, K. Maruhn, and W. Rinow (eds.),
Enzyklopaedie der Elementarmathematik, Band I Arithmetik, VEB Deutcsher Verlag der
Wissenschaften, Berlin, 1966, pp. 346 ff.

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