DECEMBER 24, 1897. ] 
the spherical trigonometry 51 pages. Answers 
are given at the end of the book, but not in 
cases where they would detract from the value of 
the examples. 
Numerical Problems in Plane Geometry. - By J. 
G. Estint. New York, Longmans, Green & 
Co. 8vo. Pp. x+144. 
Of their own motion or in conformity with the 
unanimous recommendation of the conference 
of colleges and preparatory schools held at 
Columbia University in February, 1896, most of 
the better colleges have abolished the super- 
annuated entrance requirement of a formal ex- 
amination in arithmetic, and now prescribe in 
its stead the ability to solve numerical problems 
in plane geometry and a knowledge of the met- 
rie system and in some cases of logarithms. 
Mr. Estill’s book, which is intended to furnish 
the requisite exercise in all three subjects, con- 
tains 49 pages of problems divided into books 
corresponding to the usual arrangement of the 
geometries in more general use. These are fol- 
lowed by 52 pages of recent entrance papers of 
an unusually large number of colleges, together 
with individual problems taken from similar 
papers. A five-place table of logarithms, with 
explanations and examples, occupies the next 
388 pages; and the book concludes with the 
metric tables of weights and measures, in- 
cluding tables of English and metrical equiva- 
lents. 
The book is not intended to take the place of 
other geometries, but to be used with them. 
The problems seem to be generally well se- 
lected. The metric system is used from the 
start, a favorite habit of the author being to 
give the data in metric units and to require the 
results in English measure, or vice versa. This 
is, of course, a necessary exercise within bounds, 
but, when carried to such an extreme as here, is 
calculated to give the beginner the idea that the 
metric system is an abominable contrivance for 
reckoning in terms of incommensurable num- 
bers. Occasionally, too, the answer to a 
problem is conditioned on the degree of approxi- 
mation to which the metric and English equiva- 
lents are to be taken, and this may well produce 
a feeling of uncertainty not quite in harmony 
with the notion of geometry as an exact science. 
SCIENCE. 
95g 
Plane and Solid Analytic Geometry. By FRED- 
ERICK H. BAILEY and FREDERICK 8. Woops, 
Assistant Professors of Mathematics in the 
Massachusetts Institute of Technology. Bos- 
ton, Ginn & Co. 8vo. Pp. xii+371. 
Wholly unlike trigonometry, analytic geom-- 
etry, even in the highly restricted sense in: 
which the name is employed by the present. 
authors, admits of the widest variety of treat- 
ment. To what extent shall modern coordinate- 
systems and modern methods generally be in- 
troduced? How shall the conic sections be 
defined? How shall the general equation of 
the second degree be exploited? Shall any- 
thing be said about projective relations and 
anharmonic ratios? These and many other 
questions may be settled in the greatest variety 
of ways by the author, and whatever his de- 
cision may be, he can with skill and care pro- 
duce a highly satisfactory book. 
Professors Bailey and Woods have chosen to 
exclude the more modern apparatus. They do- 
not employ determinants or projective coordi- 
nates, or anharmonic ratios, but confine them- 
selves to the ordinary Cartesian and polar co- 
ordinates and the common methods. This 
plan has itsadvantages. Beautiful and concise- 
as the modern analytic geometry is, the be- 
ginner is perhaps not able to appreciate it at 
once. He must just become acquainted with a 
large number of new and fundamental ideas 
and practice himself long and slowly, before- 
he is really able to grasp the perfect theory at 
all. If he learns great principles prematurely 
he is apt to have only a superficial understanding 
ofthem, At least this is the opinion held by 
many teachers. 
The authors have covered the usual ground 
so far as plane geometry is concerned. After 
elementary chapters on coordinates, loci, the 
straight line, and transformation of coordi-- 
nates follows one on the circle. The latter serves- 
as an introduction to the conic sections, which 
are discussed in the next chapter on the basis. 
of Arbogast’s definition. This chapter contains 
also an innovation, the discussion of the general 
equation of the second degree with the zy term. 
missing, a step which greatly unifies the follow- 
ing treatment of tangents, normals and polars. 
The discussion of the complete general equa-- 
