518 
NATURE 
[ Sept. 30, 1886 
theme of his address to the Midland Institute at Birming- 
ham, and, we have no doubt, charmed his audience with 
his manner and ready wit. But what is there in either of 
the lectures just mentioned that throws the slightest light 
on the real difficulties of the subject? Why was not Sir 
F. Bramwell moved to read a paper at the meeting 
of the British Association which took place a few 
weeks ago in the very building in which he addressed 
the Midland Institute? Had such a paper been read, 
a most interesting and animated discussion would have 
arisen, and the nation would have had the advantage of 
learning the opinions of men who have devoted their 
lives to working out the problems connected with the 
theory as well as the practice of the construction of 
ordnance. We are driven to the conclusion that the 
Ordnance Committee considered that such a course would 
have been dangerous to their self-respect ; the deplorable 
ignorance which characterises all attempts at working 
out difficult questions by Committees would have been 
too glaringly exposed ; it was much safer not to subject 
themselves to cross-examination. 
If we are wrong, if the Committee have complete 
answers to the questions raised, it is open to them to 
convince the nation of the fact, because the sessions of 
the Institution of Civil Engineers will very soon com- 
mence, and some member of the Ordnance Committee 
should be deputed to read, not a popular lecture, but a 
serious paper that will demonstrate to a roomful of 
practical engineers that our guns have been constructed 
according to the rules which guide engineers in their 
ordinary work, and that yet they have failed. We have 
small hopes that the challenge will be accepted. 
THE MATHEMATICAL AND PHYSICAL 
SCIENCES. 
fistoire des Sciences Mathématiques et Physigues. Par 
M. Maximilien Marie. Tomes 1.-IX. (Paris: Gauthier- 
Villars, 1883-86.) 
MARIE’S great work advances rapidly towards 
* completion. The first three volumes appeared 
in 1883 ; the concluding three are in the press; we have 
now before us nine volumes, bringing down the narrative 
from the time of Thales to the time of Laplace. The 
undertaking is a vast one ; and we are not surprised to 
hear that it has cost forty years of preparation. The 
learned world is to be congratulated that it has fallen 
into such able hands. M. Marie combines, in a rare 
degree, scientific with literary qualifications. A certain 
grace and poignancy of style set off his wide erudition 
and practical acquaintance with methods of teaching. 
He can be vivacious even over processes of integration. 
The accumulated mass of his materials nowhere hinders 
the lightness of his tread. Keen touches of sarcasm en- 
liven his most abstruse expositions, and agreeably remind 
his readers that a sense of humour may subsist concurrently 
with a thorough mastery of the higher analysis. 
He has accordingly produced a book which, with these 
merits and some corresponding defects, only a French- 
man could have written—one eminently interesting and 
original, at once lively and profound, instructive through- 
out if occasionally one-sided, frankly displaying the pre- 
possessions of its author, and not unfrequently—as we 
is 
shall presently show—his heedlessness of historical and 
biographical accuracy. Its characteristic merit consists 
in the lucid interpretations contained in it of the older 
methods of mathematical research. The works of ancient 
and medieval geometers are analysed, not barely in the 
view of exhibiting the results attained by them, but with 
the further purpose, most completely realised, of render- 
ing their various artifices and modes of working intelli- 
gible to the least skilled in the archeology of mathematics. 
M. Marie’s is indeed in no sense a book for beginners. 
It presupposes a considerable acquaintance with the most 
recent developments of analysis. The reader thus pro- 
vided may, however, follow with ease and pleasure the 
steps by which earlier inquirers advanced ; he may enter 
into their conceptions, place himself at their precise point 
of view, and while marvelling at the ingenuity which 
carried them so far, study the limitations of thought 
which hindered them from proceeding any farther. He 
may learn how the singular deficiency in the idea of 
abstract number which hampered the workings of such 
luminous and powerful minds as those of Archimedes, 
Apollonius, and Euclid, was supplied from the far East ; 
how Hindu algebra and arithmetic formed the comple- 
ment of Greek geometry ; how both were transmitted 
through Arabic channels to Italy, and together constituted 
the starting-point of modern discoveries. Nor is it less 
curious to watch the gradual emergence of ideas big with 
the progress of the future, such as those of negative and 
imaginary, or infinitely small quantities ; how they pre- 
sented themselves with hesitation, and were at first 
shunned and distrusted; how they grew bolder and 
insisted on recognition ; how their tentative and partial 
treatment became widened and generalised until they 
finally developed the whole extent of their capabilities. 
It is well known that Archimedes gave the first approxi- 
mation to the value of 7 ; but the occasion and significance 
of the step are often lost sight of. It marked, with the 
almost simultaneous attempt of Aristarchus of Samos to 
measure the relative distances from the earth of the sun 
and moon, the introduction of numerical calculation into 
theoretical researches (Marie, t. i. p. 59). The novel 
effort was prompted, in each case, by the interest of a 
special problem. Archimedes, naively enough, sought to 
prove that the idea of infinity had its root in enumerative 
impotence, and could be abolished by expanding the 
resources of arithmetic. He exemplified his contention 
by computing the number of grains of sand contained in 
a sphere with the interval from earth to sun for its radius. 
But a preliminary valuation of the ratio between the 
circumference of a circle and its diameter was indis- 
pensable ; and the tract on the “Dimension of the 
Circle” was accordingly, in M. Marie’s plausible view, 
written as a kind of preface to the “ Arenarius.” Inci- 
dentally to the calculations in the latter treatise, he per- 
fected the Greek system of numeration, and foreshadowed 
the principle of logarithms. 
M. Marie has ventured a kind of restoration of the 
“algebra” of Archimedes (t. i. p. 262). His remarks on 
this disputed subject are of great interest. He holds it 
impossible that his inventions should have been reached 
by the arduous path of his demonstrations ; and ascribes 
to him, accordingly, the possession of a compendious 
method of reasoning founded on the transformation of 
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REE AEE I FE Re 
