758 Proceedings of Royal Society of Edinburgh. [sess. 
Retrospect of the Period, 1813-1841. 
The characteristics of this period are best brought out by com- 
parison with those of the preceding period, it being carefully borne 
in mind, in making the comparison, that the two are markedly 
unequal in length, the period of pioneering, as we may term it, 
extending to 120 years, and the next to only about 30. 
In the first place, then, the evidence shows that as time went on 
there was considerable increase of interest in the subject, and a 
more widely spread knowledge of it; for, whereas to the longer 
period there belong 20 papers by 13 writers, for the shorter period 
the corresponding numbers are 35 and 18. Among the 18 writers, 
too, are represented nationalities which had previously not put in 
an appearance, viz., English, Italian, and Polish. 
In the second place, we have proof that the early period was by 
far the more fruitful in original results. The pioneers had mapped 
out most of the prominent features of the new country; their 
successors had consequently to concern themselves in a considerable 
degree with filling in the details. During the second period one 
finds the fundamental propositions of the first period reproduced in 
new varieties of form ; also, there are not awanting new proofs, 
extensions, and specialisations of old theorems; but of absolutely 
fresh departures there are comparatively few. An examination of 
the results numbered xlv.-lviii. will show the character of these 
departures. It will be seen that they are due to Desnanot, Scherk, 
Schweins, Jacobi, Sylvester, and Cauchy. The most notable name 
of the period is Jacobi’s, and next to it that of Schweins. There is 
no one name, however, which stands out in this period so con- 
spicuously as Cauchy’s does in the first period. Sylvester, unlike 
the others, it must be remembered, was only beginning his career, 
and we have yet to see him in the fulness of his power. 
In the next place, the second period contrasts with the first in 
that during it important work was done on the subject of special 
forms of determinants. Here, again, the noteworthy names are 
those of Jacobi and Schweins. 
Lastly, it having been noted in the retrospect of the first period 
that the subject of determinants was almost entirely a creation of 
the French intellect, we must not fail to take cognisance now of the 
