314 
is one on Jacobians, in which the elementary properties 
of these functions are put before the reader. 
In Part II., under Frullani’s theorem, reference is 
made to the recent extensions by Messrs. Elliott and 
Leudesdorf, and the demonstration by M. Zolotareff, 
that the remainder in Lagrange’s theorem is expressible 
as a definite integral is given. Our Author was not 
aware that Mr. Emory McClintock had applied the same 
mode of demonstration to the similar case of Laplace’s 
theorem, as the number of the American Fournal of 
Mathematics (vol. iv., No. 1) has reached this country 
subsequently to the publication of his article. Due im- 
portance is assigned to Definite Integrals and to the 
hyperbolic functions which now play an important part in 
investigations. 
The last 79 articles are devoted to the subject of elliptic 
functions, upon which we have written above. The 
whole of this discussion convinces one how well fitted the 
writer is to bring out a separate treatise on this branch, 
and we hope that a rumour which has reached us will 
soon become an actual fact. We may note the appear- 
ance within a very recent date of a new edition of Abel’s 
works. Passing now from the fractical treatment to the 
historical, which rightly is put in the forefront of the com- 
munication, we need not dwell in any detail upon the 
early history of the subject. That has reached its position 
of equilibrium, and no stormy winds of controversy are 
likely to disturb its calm. But we did wonder what posi- 
tion Mr. Williamson would take when he came to discuss 
the commercium epistolicum business. We ourselves 
had long thought that little could be added to De Mor- 
gan’s summing-up, still we remembered some strong lan- 
guage which once appeared in our epistolary columns (see 
vol. v. pp. 62, 81, 121). We need hardly say that Mr, 
Williamson appears to have gone over all the sources of 
information which we had ourselves recently examined, 
and he seems to us to have come to the right solution. 
We append his summing-up :— 
“Tt is the less necessary now-a-days to.enter into the 
merits of this great quarrel, inasmuch as it has long been 
agreed upon, by all mathematicians who have examined 
into the controversy ; that Newton and Leibnitz are both 
justly entitled to be recognised as independent discoverers 
of the principles of the calculus, and that, while Newton 
was certainly master of the method of fluxions before 
Leibnitz discovered his method, yet Leibnitz had several 
years priority of publication.”’ 
A VERY effective article is the one on insects by Mr. R. 
McLachlan, F.R.S. In it we find condensed into a few pages 
a very correct and instructive history of this immense class 
of that subdivision of the animal kingdom now known as 
the Arthropods. As in most of the classes of the animal 
kingdom, the limits of the class Insecta are not very 
sharply defined. In general every one thinks that it is an 
easy task to define an insect, but to do so in the way 
that we can define a mammal is not possible with our pre- 
sent knowledge; still a fairly satisfactory diagnosis is 
here given to us, and the certain small aberrant groups 
are not altogether overlooked. The insects not only form 
the largest class of the arthropodous sub-kingdom, but, 
according to the author, possibly outnumber all the other 
members of the animal kingdom besides. It seems cer- 
tain that at the present time there are about 80,000 pre- 
NATURE 
[ Feb. 
sumably distinct species of beetles described, and it seems 
safe to assume that the number of known species of other 
orders is greater, thus giving a tota] of about 200,000. 
But as yet we are only on the threshold of our knowledge 
of the forms that actually exist in Nature—many enor- 
mous groups of living forms being still only very partially 
studied—so that we may pretty confidently anticipate 
that some day, not very far distant, the number of known 
forms will not fall far short, the author writes, of 
1,000,000, but we would feel inclined to write instead 
500,000, As to the antiquity of the group, indications of it 
appear in the Devonian series, and become more marked 
and numerous in the Carboniferous. With few exceptions 
these are those of insects belonging to orders in which the 
metamorphoses are incomplete, and there seems no evi- 
dence that any anthrophilous insects, such as bees or 
butterflies, were then in existence. As one ascends the 
geological scale, insect life gradually develops itself, not 
becoming however at all abundant until Tertiary times. As 
to their geographical distribution, they would seem to be 
2, 1882 
everywhere on the earth's surface, the more attractive © 
forms are more often denizens of the tropics, but showy 
butterflies were found during the late Arctic Expedition — 
almost up to 83° N. lat. They occur on land, in fresh- 
water, in hot springs, in brine, in the deep recesses of 
vast caverns, on the surface of the ocean, and we may add, 
are at home under the flow and ebb of its waters all around 
our own shores. The subjects of the duration of their 
life, their uses to man, the injuries they directly and in- 
directly inflict on him, their parasitism, their anatomical 
structure, the wondrous story of their change of form, 
these are all ably though briefly handled. The systematic 
portion of the article, though of necessity condensed into 
the smallest possible compass, is especially interesting. 
The sequence of the orders is :— 
( IHym2noptera 
Coleoptera 
Diptera 
Lepidoptera 
Neuroptera 
Orthoptera 
** | Hemiptera 
Collembola 
“* | Thysanura, 
Metamorphosis complete ... 
Metamorphosis incomplete 
No metamorphosis 
The stumbling-block of all systematists has been the 
Linnean order Neuroptera. The author manages it after 
the method of Erichson, placing those neuropterous forms 
with an incomplete metamorphosis as the sub-order Or- 
thoptera. He tucks in the Diptera between the Coleoptera 
and Lepidoptera, a position we doubt very much thatthey 
will consent to occupy, though on the author’s views as to 
the absolute value of metamorphosis in classification, this 
position might stand. The Mallophaga are without hesi- 
tation placed as degraded Pseudo-neuroptera, and the 
Anopleura as equally degraded Hemiptera, despite what 
veterans in entomological science (ze. people too old to 
take in new ideas) may say. But, says Mr. McLachlan, 
there is a breaking point to elasticity even in ideas of 
classification; and so he avails himself of an assertion of 
Sir J. Lubbock, that the Collembola and Thysanura— 
sprightly things—are scarcely within (we note not with- 
out) the pale of true insects. Still fearing that no one 
else might write about them, that they might be over- 
looked by the writers of the articles on Crustacea, Arach- 
