Entered, according to Act of Congress, in the year one thousand eight hundred and sixty - six, by

HARPER & BROTHERS,

in the Clerk's Office of the District Court of the Southern District of New York.

ΤΟ

HORACE WEBSTER,

L L. D.,

PRESIDENT OF THE COLLEGE OF THE CITY OF NEW YORK,

AS A TRIBUTE OF RESPECT FOR THE FAITHFUL PERFORMANCE OF THE DUTIES BELONGING TO HIS OFFICE,

A TOKEN OF SINCERE REGARD FOR HIS DEVOTION TO THE CAUSE OF SOUND EDUCATION, AND

A MEMORIAL OF PERSONAL FRIENDSHIP CEMENTED BY A DAILY INTERCOURSE OF MANY YEARS,

This Volume

IS MOST SINCERELY DEDICATED BY

THE

AUTHOR.

PREFACE.

Ir has been my endeavor, in this volume, to furnish the student with every aid which can be afforded consistently with securing the end which mathematical studies are designed to attain. I have not attempted to lessen the amount of thought and attention which are necessary for him to put forth, but only to prevent him from wasting that thought and

attention.

The most perfect demonstration of a mathematical theorem is that which, while rigorously exact, is expressed in the fewest and simplest words. The "Elements of Euclid," with some exceptions, are acknowledged to be absolutely perfect in this respect. Taking these for a basis and model, I have reduced the essential features of Geometry to so small a compass that I have been enabled to bring that science, as well as its cognate and dependent ones, Trigonometry, Mensuration, Surveying, and Navigation, within the compass of a single volume.

In the chapters on Trigonometry, Mensuration, Surveying, and Navigation, the student will find much that will, by exercising his analytical powers, fit him for the successful study of the higher branches of Mathematics.

I have prepared this treatise with a special view to

meet the wants of students in the College of the City of New York, who are early trained to understand and appreciate the subtle methods of demonstration occasionally used in it.

The great success which has followed the introduction of my Algebra, Analytics, and Differential and Integral Calculus, is some guarantee for the value of this volume; and the high standing which the students of this free college have taken among engineers, architects, and educators of our country, furnishes also a living commentary on the method of instruction which is employed therein.

GERARDUS BEEKMAN DOCHARTY.

College of the City of New York, July, 1866.