s words
common to two at least of the three principal groups of
Romance:--Italian, Spanish and Portuguese, and Provencal and French.
The Italian, as nearest the original, is placed at the head of each
article. The second part treats of words peculiar to one group. There
is no separate glossary of Wallachian.

Of the introduction to the grammar there is an English translation by
C. B. Cayley. The dictionary has been published in a remodelled form
for English readers by T. C. Donkin.

DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau,
romantically situated in the deep valley of the Lahn, here crossed by an
old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500.
It is overlooked by a former castle of the counts of Nassau-Dillenburg,
now a prison. Close by, on an eminence above the river, lies the castle
of Oranienstein, formerly a Benedictine nunnery and now a cadet school,
with beautiful gardens. There are a Roman Catholic and two Evangelical
churches. The new part of the town is well built and contains numerous
pretty villa residences. In addition to extensive iron-works there are
sawmills and tanneries. In the vicinity are Fachingen, celebrated for
its mineral waters, and the majestic castle of Schaumburg belonging to
the prince of Waldeck-Pyrmont.

DIFFERENCES, CALCULUS OF (_Theory of Finite Differences_), that branch
of mathematics which deals with the successive differences of the terms
of a series.

1. The most important of the cases to which mathematical methods can be
applied are those in which the terms of the series are the values, taken
at stated intervals (regular or irregular), of a continuously varying
quantity. In these cases the formulae of finite differences enable
certain quantities, whose exact value depends on the law of variation
(i.e. the law which governs the relative magnitude of these terms) to be
calculated, often with great accuracy, from the given terms of the
series, without explicit reference to the law of variation itself. The
methods used may be extended to cases where the series is a double
series (series of double entry), i.e. where the value of each term
depends on the values of a pair of other quantities.

2. The _first differences_ of a series are obtained by subtracting from
each term the term immediately preceding it. If these are treated as
terms of a new series, the first differences of this series are the
_second di