Oct. 7, 1886] 
_ NATORE 
4 
by the multiplication of labour-saving machinery, should be dis- 
tinguished by an unexampled development of this most refined 
and most beautiful of machines. That such has been the case, 
no one will question. The improvement has been in every part. 
Even to enumerate the principal lines of advance would be a 
task for any one—for me, an impossibility. But if we should 
ask in what direction the advance has been made, what is to 
characterise the development of algebra in our day, we may, I 
think, point to that broadening of its fields and methods which 
gives us ‘multiple algebra.’ ” 
The speaker then gave a critical historical review of the dif- 
ferent contributions of Hamilton, Mobius, Grassmann, Saint- 
Venant, Cauchy, Cayley, Hankel, the Peirces, father and son, 
and Sylvester, to these new methods of mathematical analysis, 
showing the additions and developments made by each to the 
various subjects. 
In the second part of the paper Prof. Gibbs criticised the 
methods of some modern writers on these subjects, showing how 
they failed to grasp the full significance and bearings of the 
matters they were dealing with, being too much hampered by 
the old ideas and methods of simple algebra. 
In the third part of his paper Prof. Gibbs directed attention 
more critically to multiple algebra itself, and inquired into its 
essential character and its most important principles. 
Then followed along discussion of the fundamental concep- 
tions and methods of modern mathematics, which nothing but 
publication in full could render intelligible, and that only to 
mathematicians. 
The fourth part of the paper was devoted to consideration of 
some of the applications of multiple algebra. From this we 
quote the following :—‘‘ First of all, geometry, and the geo- 
metrical sciences which treat of things having position in space, 
—kinematics, mechanics, astronomy, crystallography,—seem to 
demand a method of this kind, for position in space is essentially 
a multiple quantity, and can only be represented by simple 
quantities in an arbitrary and cumbersome manner. For this 
reason, and because our spatial intuitions are more developed 
than those of any other class of mathematical relations, these 
subjects are especially adapted to introduce the student to the 
methods of multiple algebra, Here Nature herself takes us by 
the hand, and leads us along by easy steps, as a mother teaches 
her child to walk. In the contemplation of these subjects 
Mobius, Hamilton, and Grassmann formed their algebras, 
although the philosophical mind of the last was not satis- 
fied until he had produced a system unfettered by any spatial 
relations. It is probably in connection with these subjects 
that the notions of multiple algebra are most widely dis- 
seminated. Maxwell’s ‘Treatise on Electricity and Mag- 
netism’ has done so much to familiarise students of physics 
with quaternion notations, that it seems impossible that this 
subject should ever again be entirely divorced from the methods 
of multiple algebra. I wish that I could say as much of astro- 
nomy. It is, I think, to be regretted that the oldest of the 
scientific applications of mathematics, the most dignified, the 
most conservative, should keep so far aloof from the youngest of 
mathematical methods ; and standing, as I do to-day, by some 
chance, among astronomers, although not of the guild, I cannot 
but endeavour to improve the opportunity by expressing my con- 
viction of the advantages which astronomers might gain by em- 
ploying some of the methods of multiple algebra. A very few 
of the fundamental notions of a vector analysis, the addition of 
vectors and what quaternionists-would call ‘the scalar part and 
the vector part of the product of two vectors’ (which may be 
defined without the definition of the quaternion)—these three 
notions, with some four fundamental properties relating to them, 
are sufficient to reduce enormously the labour of mastering such 
subjects as the elementary theory of orbits, the determination of 
an orbit from three observations, the differential equations which 
are used in determining the best orbit from an indefinite number 
of observations by the method of least squares, or those which 
give the perturbations when the elements are treated as variable. 
In all these subjects the analytical work is greatly simplified, 
and it is far easier to get the best form for numerical calculation 
than in the use of the ordinary analysis.” 
Then followed illustrations of the various methods of applying 
multiple algebra to different classes of problems. 
Prof. Brackett’s address on ‘‘ The Seat of the Electromotive 
Force” was essentially a 7éswmé of the history of the investiga- 
tions to find the source of the current in galvanic batteries. No 
attempt was made to settle the question, which has been so long 
a bone of contention. 
In his address to the Section of Biology, Dr. H. P. Bowditch, 
of Boston, concluded that investigations into the chemical 
changes, the heat production, and the fatigue of active nerves, all 
tend to results more favourable to a kinetic than toa discharging 
theory of nerve action. 
In the Section of Anthropology a novel and ingenious method 
of getting an insight into the unconscious mechanism of author- 
ship was described by Mr. T. C. Mendenhall, under the title 
“*Characteristic Curves of Composition.” The method consists 
in counting the number of words of each length, from one letter 
to fourteen, fifteen, or as long as were found, and plotting the 
result on a curve, in which the abscissee represented the number 
of letters in the word, and the ordinates the number of words 
per thousand of each length. It was shown that while the curve 
resulting from each thousand words was not entirely regular, that 
resulting from five thousand was much more regular, and that 
from ten thousand almost entirely so. The inference from this 
was, that the phenomenon which the curve represented was a 
regular one, and that it was an expression of the peculiar yoca- 
bulary of the author. Moreover, by comparing the respective 
curves, one would be able to judge whether two works were 
written by the same author, and perhaps even decide the con- 
troversy whether Bacon wrote Shakespeare. Mr. Mendenhall’s 
method was to count a thousand words at a sitting, and then 
turn to another part of the book. One soon acquired the art of 
counting at a glance the number of letters in each word, and, 
with an assistant to record the result, one thousand words could 
be counted in a half-hour. Curves derived from Dickens 
(‘*Oliver Twist”) and Thackeray (‘‘ Vanity Fair”) were re- 
markably similar, thus suggesting that the subject-matter might 
cause the peculiarity of the curve, while those from John Stuart 
Mill (‘‘ Political Economy” and ‘‘ Essay on Liberty”) differed 
from them in having more long words and fewer short ones, 
though words of two letters (prepositions mainly) were most 
abundant in Mill. The average length of the novelists’ words 
was 4°38, and that of the philosopher 4°8. 
The geological interest of the meeting at Buffalo naturally 
centred in the excursion to and discussion of the Falls and gorge 
of Niagara. Dr. Pohlman, of Buffalo, described the district to. 
be visited on Saturday, and called particular attention to the 
occurrence of drift-filled antecedent channels on the line selected 
by the post-Glacial overflow of Lake Erie, which would gradually 
diminish the amount of rock-cutting required in the excavation 
of the present gorge, and thus reduce the time since the over- 
flow began. The geological members of the excursion party 
therefore gave close attention to these matters, and, as a whole, 
regarded the heavy drift between the sloping rocky banks at the 
whirlpool, and the wide, open valley, with its plentiful drift at 
St. David’s, as sufficient evidence of an old buried channel con- 
necting these points, and probably heading up above the whirl- 
pool towards the bridges. But there seemed no sufficient reasoa 
for any confident belief in a branching old valley from the 
whirlpool towards the Lewiston bluffs: in making this lower 
part of the gorge there must have been a long period of deep- 
rock-cutting between the first leap of the Falls over the bluff and 
the time of their discovering the old drift-channel and the whirl- 
pool. The estimate of the age of the Falls was presented by 
Messrs. Woodward and Gilbert, of the Geological Survey, and 
their remarks greatly interested a large audience that had 
gathered on the announcement of the discussion, Mr. Wood- 
ward had just completed a survey of the Horseshoe Falls, and 
by comparing his results with those of the State Survey in 1842, 
and of the Lake Survey in 1875, he found an average recession 
for the whole face of the Fall of about 2;4; feet per annum ; but 
as the central parts of the curve, where the water is deepest, has 
retreated from 200 to 275 feet in the eleven years since 1875, an 
average retreat of 5 feet per annum does not seem at all im- 
probable. Mr. Gilbert then discussed the beginning of the 
Falls as controlled by the drainage of the lakes. When the 
retreating ice-sheet stood so as to obstruct the St. Lawrence 
and Mohawk drainage channels to the east, a broad sheet of 
water, representing a confluent of Erie and Ontario, stood at a 
high level over the present Niagara limestone plateau, and 
probably drained south-westward to the Ohio. When further 
melting opened the Mohawk Channel, the great double lake fell 
to a lower level, and was separated into its two members, 
Ontario sinking to the level of its outlet at Rome in Central 
New York, but Erie being held higher by the rim of the 
Niagara plateau. This was the birth of the river and the Falls, 
and since then they have been at work on the gorge. The age 
of the falls thus carries us ba:k to a tolerably definite point in 
