The theory of determinants in the historical order of development, by Sir Thomas Muir.

406 HISTORY OF THE THEORY OF DETERMINANTS

The rest of the paper (pp. 13-24) is occupied mainly with continued-fractions, and contains nothing new in regard to the related
determinants. Possibly the nearest approach to a fresh result is
the use of the known identity

a0-al+a2-...

- ao
= + ao a, a
a1 ao- a, + aa
al —a,+.

to suggest (p. 16) that
ao-a1 -1...... = aa2.... an,.{a0-a +a2-a3 +....},
aoa2 a-a2 - 1....
ala a -a...
an equality easily established otherwise.

GUNTHER, S. (1873).
[Darstellung der Naherungswerthe von Kettenbriichen in independenter Form. iv+128 pp. Erlangen.]
When we come to Giinther we have reached the first formal
treatise on our subject. His booklet consists of three chapters,
the second of which, with the title 'Darstellung der Zahler und
Nenner jedes Naherungsbruches in Determinanten-Form,' is that
which mainly concerns us. The first chapter (pp. 1-30) is of the
nature of a historical review of the various modes proposed for the
calculation of convergents, and the third is professedly an application of the results of the second chapter to 'Analysis, Algebra,
und Physik.'
That portion (~ 6) of chapter i. which deals with the introduction
of determinants into the treatment of continued-fractions attributes
the first idea of the existence of the relation to Ramus (1855); notes
that Heine (1859) discovered it independently; and nevertheless
represents Heine as having used a result of Painvin's. All this,
of course, stood at the time* in need of serious modification, which
unfortunately was not forthcoming, with the result that some of
*For Heine's repudiation see his Handbuch der Kugelfunction, i. (1878),
pp. 261-262.