FEBRUARY 23, 1912] 
mathematics regarded as “the universum of 
exact thought,” rather than to algebra and 
geometry conceived as special branches there- 
of. Basic concepts and central concepts, 
though the two categories may intersect, are 
in general not the same, else the book would 
doubtless have admitted to a prominent place 
the notion of invariance, a notion that, be- 
sides being of central importance in mathe- 
matics, serves to ally the interests of this 
science with those of science in general, and 
with those of philosophy, theology, religion 
and art. It is the chief unifier of the great 
forms of human interests and endeavor. Pro- 
fessor Young has admirably shown that any 
science must contain two sorts of ideas, the 
assumed and the defined; besides these it uses 
ideas that it does not contain (as subject 
matter); the like is true of propositions; 
somebody ought, in respect of some science, 
to assail the problem of indicating those ideas 
that are used by the science without being a 
part of its content. In the chapter dealing 
with “the logical significance of definitions ” 
Professor Young might well have said, what 
he doubtless knows, that, for example, the 
notion of definition is no part of the subject 
matter of logic or mathematics, though for 
the sake of convenience the notion is contin- 
ually there in use. The conception of mathe- 
matics presented in italics on page 221 leaves 
the science without any unity except such as 
belongs to a mere collection, which can never 
satisfy. The defect is partly cured on page 
925. In a democracy it is a duty of scholars 
to render scientific concepts intelligible to the 
public intelligence, and Professor Young’s 
book is a valuable contribution to such high 
service. 
The descriptive geometry treated by Pro- 
fessor Wilson and Professor Bartlett is not 
to be confounded with that great variety of 
geometry, called descriptive by Russell and 
others, which was founded by Pasch in his 
“ Vorlesungen tiber neuere Geometrie ” (1882) 
and a few years later cast in symbolic form 
by Peano, but it is that branch of geometry 
which has for its object the representation of 
SCIENCE 
305 
3-dimensional figures by means of their pro- 
jections upon two or more planes, a method 
invented by Gaspard Monge (a peddler’s son) 
in response to military exigencies in France 
and set forth by him in 1800 in his “ Géomé- 
trie descriptive.” Professor Wilson’s aim, 
“to present a sound theoretical treatment ” 
and not to win the student by means of mere 
appeal to “short cuts” and his “ practical ” 
interest, is laudable. As to the extent to 
which the end has been attained, some mathe- 
maticians may be disappointed in not finding 
here a system of postulates. On the same 
page (2) we are told that parallel lines are 
said to have two points in common at infinity, 
that parallel planes meet each other in a 
common line at infinity, and that descriptive 
geometry and perspective are a part of pro- 
jective geometry. The traditional use of 
“line” for curve is adopted. Two consecu- 
tive points are regarded as points “ infinitely 
close” together but not coincident (p. 81) 
and as coincident points (p. 93). The state- 
ment that “the secant approaches tangency 
and become such when the two points [of 
secancy] become coincident” will serve (even 
after typographical correction) to exemplify 
a not infrequent occurrence in the book of 
unprecise statement. One can not but feel 
that so good a book ought to be better. ‘ypo- 
graphically and mechanically it is pleasing; 
there is appended a goodly list of exercises; 
and the point of view is somewhat more gen- 
eral than is common in American text-books 
in this subject. 
When will some breath of modern mathe- 
matics get into our text-books of descriptive 
geometry? Church’s “Elements,” published 
in 1864, has reigned, partly because of its 
merits, for nearly five decades. To meet the 
new demands in respect of matter and of 
presentation, Professor Bartlett has not 
deemed it necessary to depart essentially 
from the venerable text of Church. Conse- 
quently this interesting and instructive new 
volume has the scientific odor of a geometric 
past, despite the excellence of the drawings 
and pictorial representations of certain more 
