Marcu 21, 1907 | 
NA TORE 
483 
metric logarithms is one which shows clearly the | they bore the heading ‘‘ Things that Ought Not to 
dual use of the tables for reading off the logarithmic 
sines of angles and the logarithmic cosines of. their 
complements. With small books of tables this is best 
done by using the right-hand column ‘and top line 
for logarithmic sines, the left-hand column and 
bottom line for logarithmic cosines. This is the 
arrangement adopted by Dr, Briggs in his ‘‘ Clive’s 
Mathematical Tables.’’ The same arrangement is 
followed in regard to tangents and co-tangents. The 
tables of secants and cosecants are another desirable 
feature. By adding the logarithm of a cosecant in- 
stead .of subtracting the logarithm of a sine, many 
compound expressions may be calculated by a single 
addition sum. It is a pity that logarithms of re- 
ciprocals are not also given. The tables are given to 
five places, and corrections are given in all of them 
where the differences are irregular. The explanatory 
matter is very useful to students, notably the defini- 
tion of significant figures. 
Dr. Denning, in his introduction, remarks that 
‘Criticisms and suggestions for future editions wil! 
be welcomed.’’ The first criticism which suggests 
itself is that a book where logarithms of numbers 
less than four have necessarily to be taken from a 
table of antilogarithms, and logarithms of numbers 
greater than four from a table of logarithms, is far 
too ingenious to put into the hands of a beginner. 
The object of this arrangement is, of course, to 
avoid the large and irregular differences that occur 
with logarithms of the lower numbers and _ anti- 
logarithms of the higher ones. If the book is not 
meant for beginners the arrangement is good, but 
for teaching the use of tables the complete tables of 
logs. and antilogs. should be given, and students 
should be taught later on when to use each. The 
insertion of corresponding tables for obtaining 
logarithms of reciprocals is a good feature. It seems 
rather curious that no one has adopted the plan of 
bordering a table of antilogarithms with a bottom 
line and right-hand column containing the arith- 
metical complements of the numbers in the top line 
and left-hand column. Such an antilogarithm table 
would give logarithms of reciprocals very simply. 
The arrangement of the trigonometrical tables is 
not very clear. There are no head- or footlines to 
the middle page, and while the columns look to run 
on from one page to the next, they do not really do 
so. The left-hand column of the first two pages goes 
from 0° to 15°, and we naturally expect to find 15° 
to 30° on the next page, but instead of that we find 
30° to 45°, the entries for 15° to 30° being on the 
right-hand column of the first two pages. The mis- 
print “ co-functions ’? at the foot of p. 16 does not 
really introduce additional confusion. The book con- 
tains tables of squares and cubes for those who like 
to indulge in such luxuries. Pages of physical and 
chemical constants, electric units and data, together 
with some of the differentiation and integration 
formulae also given, are really useful, and, finally, 
some “‘simpler mechanical relationships *’ and state- 
ments of the binomial and Maclaurin’s theorems 
would be of greater value to the average student if 
NO. 1951, VOL. 75] 
be Learnt.” 
These criticisms do not preclude us from stating 
that the tables will be very useful to such 
students as have learnt to find their way about in 
them. 
science: 
GEOGRAPHY AS A LIVING SCIENCE. 
Beobachtung als Grundlage der Geographie. By 
Prof. Albrecht Penck. Pp. 63. (Berlin: Gebruder 
Borntraeger, 1906.) Price 1.60 marks. 
HIS little work, which is choicely printed, is a 
record of a delightful personality. It contains 
the parting address of Prof. Penck to the students 
of Vienna, and his introduction to those of Berlin, 
now the suzerain-city of the land where he was born. 
The first words, ‘‘ Liebe Freunde,’’ ring very truly 
in our ears, and the title of the pamphlet recalls to 
those friends scenes in very many lands. Especially 
prized by the present writer is a little photograph— 
a mere imperfect sketch, if you will—in which Prof. 
Penck is seen writing up his notes in the open air, 
on the very edge of one of the world’s great land- 
scapes, where the scarp of the African tableland 
goes suddenly down towards the sea. Like his dis- 
tinguished botanical colleague, Prof. Engler, Penck 
has realised the tradition of Humboldt, and has felt 
that the German people “‘ darf sich in geographischer 
Arbeit nicht auf sein Gebiet beschranken, és muss 
solche auf der ganzen Erde leisten’’ (p. 60). 
The striking contrast of geographical position makes 
it necessary to urge the claims of travel more 
strongly in Berlin than in Vienna. The romance 
of Vindobona and Carnuntum, of the Germanised 
city facing the great ‘‘ Kessel, in den sich Volker- 
woge auf V6lkerwoge stiirzte,’’ calls us eastward in 
the first few pages, and we ask What 
has Berlin to offer after this? In the last pages, 
however, we meet our anSwer—Germany centres in 
the flat land of Berlin, but Germany has spread her 
wings. Near the North Pole lies King William Land, 
near the South Pole lies Emperor William IT. Land, 
and the union of the German States has allowed 
all Germany to look towards the sea. On _ this 
medium, which no longer divides but joins the con- 
tinents, we trust that ships may bear in all directions 
the students of Berlin, imbued as they cannot fail 
to be with the high and genial spirit of their master. 
In the Austrian section of the pamphlet, Prof. 
Penck shows how tectonic geography has specially 
developed in Vienna. He urges, however, that the 
relations between internal structure and surface- 
features are not always so close as has been sup- 
posed. The forms associated with the higher regions 
of the Alps are thus due less to the recent folding of 
the chain than to the surface-action of the glaciers 
of the Ice age and of modern times (p. 16), which 
continuously carry away, by a nibbling action, frag- 
ments from the valley-walls. The author believes that 
the Alps were far more rounded before the advent’ 
ourselves, 
| of the Ice age, though they possessed (p. 20) a much: 
