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Throughout the world what an expansion! till to-day, through the 
influence of their technical institutions of all classes, the civilized nations 
are battling for the industrial supremacy of the earth. 
So essential is modern geometry to technology that it is safe to say 
the latter can not exist without the former. Through geometry the con- 
trolling mind translates the creative idea to the willing worker, and what 
was only a dream of beauty or utility now stands clothed in material form 
under the eye of the world for its edification, elevation and use. The 
artist whose masterpieces adorn our walls, first groups his figures as he 
wishes them to appear, then he calls geometry to his assistance to make 
them seem to be where he wishes them. 
We mistake, however, if we confine the services of geometry to tech- 
nology. Nature is continually inviting the observant mind to geometric 
study. The beautiful crystalline forms which abound in the rocks of the 
globe, in the snow and the ice speak of unity in infinite diversity. Under 
the microscope the thinnest plate of shapeless rock, the very particles of 
dust at our feet, tell through shape alone a story of origin and character 
interesting and yaluable alike to the physicist, the geologist, and the 
chemist. The latter, interested in the ultimate forms of matter, finds 
suggestions for valuable theories upon atomic forms, and constructs geo- 
metric molecules in which a dissimilar position of a characteristic atom 
will in a measure explain such curiosities in nature as right and left- 
handed sugar, which, though having different properties, still are made of 
precisely the same constituents in the same proportions. 
Analytic geometry occupies the border land between geometry and 
the higher analysis. The elements of this subject so far as the construc- 
tion of loci is concerned are rapidly becoming the possession of the read- 
ing public. The variations of temperature, of humidity, of productivity, 
of commerce, of population, of crime and of the price of wheat, from 
hour to hour, or from day to day, or from season to season, are imme- 
diately expressed to the eye by curves in which portions of time are the 
horizontal measurements and the various values of the function are the 
vertical. In science the natural way to express one series of facts de- 
pendent on another is through a curve. Passing on from loci, which are 
so full of meaning and so suggestive of causal relations, it is customary 
to discuss in detail the circle, the parabola, the ellipse and the hyperbola. 
The laws of gravity lead to these curves, and they are the fundamental 
orbits of the bodies of the universe. In terrestrial matters they lie at the 
