## 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:
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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