itions: _A line is length without breadth_, and _A surface is
length and breadth, without thickness_. Mr. Smith asserts that
these definitions are false, and sustains his position by numerous
demonstrations in the pure Euclidean style. He declares that every
mathematical line has a definite _breadth_, which is as measurable as
its length, and that every mathematical surface has a _thickness_,
as measurable as the contents of any solid. His demonstrations, on
diagrams, seem to be eminently clear, simple, and conclusive. The
effects of this discovery and these demonstrations are, to simplify
very much the whole subject of Geometry and mathematics, and to clear
it of many obscurities and difficulties. All geometers heretofore
have claimed that there are _three kinds_ of quantity in Geometry,
different in their _natures_, and requiring units of different natures
to measure them. Mr. Smith shows that there is but _one_ kind of
quantity in Geometry, and but one kind of unit; and that lines,
surfaces, and solids are always measured by the same identical unit.

Besides the leading features of the work which we have thus briefly
described, it contains many new and beautiful demonstrations of
general principles in Geometry, to which the author was lead by his
new methods of investigation. Among these we may mention one, viz.,
"The square of the hypothenuse of a right-angled triangle equals four
times the area of the triangle, plus the square of the difference of
the other two sides." This principle has been known to mathematicians
by means of arithmetic and algebra, but has never before, we believe,
been reduced to a geometrical demonstration. The demonstration of
this principle by Mr. Smith is one of the clearest, simplest, and
most beautiful in Geometry. The work is divided into three parts,
I. The Philosophy of Geometry, II. Demonstrations in Geometry, and
III. Harmonies of Geometry. The demonstrative character of it is
occasionally enlivened by philosophical and historical observations,
which will add much to its interest with the general reader. We have
too little skill in studies of this sort to be altogether confident
in our opinion, but certainly it strikes us from an examination of the
larger and more important portion of Mr. Smith's essay, that it is an
admirable specimen of statement and demonstration, and that it must
secure to its author immediately a very high rank in mathematical
science. We shall await with much i