April 20, 1871] 
NATURE 
483 
a ee ee eee 
the objects they so grandly figure and often so elaborately 
describe. Even the reports of scientific expeditions may 
frequently be searched in vain for this kind of infor- 
mation, which has to be gleaned from authorities not 
always trustworthy, from scattered papers, or from 
books of travel such as have been issued in this 
country on the Malay Archipelago and the River 
Amazon. It is mortifying to exhibit forms distinguished 
by extraordinary developments of structure, and to be 
able to say nothing on associated habits. Such strange 
developments were once considered to be mere freaks of 
nature, but no one now doubts their having a biological 
and even a genealogical significance. What a field is 
is here opened! How little of the biology of a new 
form has been exhausted when it has been collected, 
named, described, figured, and even dissected !_ Scientific 
treatises have prepared the foundation for a solid know- 
ledge of the subject, but there would be occasion for 
regret if biology should ever come to be regarded by 
students in an aspect too exclusively histological or even 
physiological, if such a view operated to the prejudice of 
genuine out-of-door observations. The greatest advance 
in Natural History made in the present, or perhaps in any 
other generation, has been mainly accomplished by two 
observers who are pre-eminently life-historians. 
Little need be said of the miscellaneous illustrations 
contained in the upright portion of the table-cases. They 
seem to be very successful in engaging the attention of 
visitors of all classes—a point which is felt to be of pro- 
minent importance where the admissions amount to about 
2,000 daily. What brings them here? is a question which 
again and again suggests itself. Reduce the number by 
all the idlers and sight-seers who, no doubt, constitute a 
large proportion of the gathering ; still, if only 100 or 
even 50 seek some kind of instruction, even these in the 
course of a year form a large and teachable class. As a 
firm believer in the humanising effect of an intelligent 
interest in Natural Science, to myself the grand museum 
problem seems to be, how to make such an institution 
most beneficial to the greatest number. 
HENRY H. HIGGINS 


PRE-EUCLIDIAN GEOMETRY 
Die Geometrie und die Geometer vor Euklides. Von 
"Prof, C. A. Bretschneider. (Leipzig: B. G. Teubner, 
1870. London: Williams and Norgate.) 
NTIL the appearance of this book, Montucla’s 
celebrated ‘‘ History of Mathematics” contained 
almost all that was known about the early history 
of Mathematics up to the present time. Later 
historians, even the careful Chasles, have almost exclu- 
sively copied him, without taking the trouble of searching 
the Greek writings for themselves. Montucla’s remarks, 
however, are not only meagre, they are even not always 
correct. For this reason Prof. Bretschneider has collected 
all important passages in Greek writings which refer to 
the state of Geometry in Greece in the time before Euclid. 
This author is the first of whom complete works have 
reached us; with him, therefore, a History of Geometry 
begins. With regard to the ante-Euclidian times we cannot 
advance beyond conjectures, and these will always de- 
pend more or less upon the individuality of the historian, 
Perfectly aware of this, Prof. Bretschneider gives in the 
little volume before us, of about 180 pages, not merely 
his conclusions, but he adds the whole material which he 
has collected. Instead of simply referring to an author, 
he quotes 77 exfenso the original Greek text, and adds 
translations. Thus every reader is at once enabled to 
form his own opinion, which, we feel assured, will in most 
cases agree with that of our author. 
In the first section Prof. Bretschneider considers the 
Geometry of the Aigyptians, and tries to make out how 
far their knowledge extended. He protests against the old 
opinion that they possessed only the very first notions of 
geometry, and that the Greeks did not obtain anything 
from them worthy of the name of science. He refutes 
equally strongly the statement of some modern writers, 
who maintain that the Aigyptians knew not only all that 
Euclid gives in his Elements, but were even acquainted 
with the theories of quadratic equations and of conic sec- 
tions. According to him geometry originated in Agypt, 
where it was cultivated for practical purposes. It was rather 
an art of mechanical drawing than a science proper. The 
results obtained were collected in the form of fixed rules, al- 
ways ready for use, most of them probably strictly proved 
others perhaps resting onexperienceonly. Those collections 
of rules were at an early age included in the religious 
canons. Any alteration, any improvement, was thus almost 
impossible, especially as the only cultivators of Science, 
the priests, would take a secondary interest only in any- 
thing not strictly connected with religion. Thus it is not 
to be wondered at that geometry remained for thousands 
of years in the same state, till the unfettered genius of the 
Greek nation began to cultivate it, and then the progress 
was a most rapid one. 
It is, however, remarkable, although natural enough, 
that the Greeks retained to a certain extent the form into 
which AZgyptian priests had cast their propositions. 
For this fact there exists a testimony in a papyrus at the 
British Museum, formerly in the possession of the late 
Mr. Rhind, which contains a pretty complete treatise on 
Applied Mathematics, in the shape of problems which 
are stated in that peculiar form with which we are so well 
acquainted through Euclid’s Elements, Dr. Birch, who 
has given an account of it, dates it as far back as 3400— 
3200 B.C. Prof. Bretschneider traces many other pecu- 
liarities in Euclid’s Elements—for instance, the order 
of propositions—back to the same source; so that the 
‘Egyptian priests, who lived about 6000 years ago, have, 
in the most direct manner, influenced the mode of teaching 
geometry in English schools even at the present time. 
The extent of Aigyptian Geometry is estimated as 
follows :—the theory of angles and parallel lines ; the 
construction of triangles, parallelograms, and trapezoids 
from given parts, and the determination of their areas ; 
the elementary propositions of the circle together with 
the inscribed regular polygons ;—this is about the sum 
total of Plane Geometry. In Solid Geometry their know- 
ledge was limited to the first notions about lines perpen- 
dicular to a plane, and the theory of parallel lines and 
planes in space. They were acquainted with the existence 
of prisms, regular pyramids of four sides, of the right 
cone and cylinder, the sphere, and of the regular solids 
with the exception of the dodecahedron, which is the only 
one discovered by Pythagoras, Of the properties of 
