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OF

GEOMETRY, MENSURATION,

TRIGONOMETRY, LAND-SURVEYING,

AND LEVELLING:

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PRACTICAL PROBLEMS IN ESTIMATION, SURVEYING,
AND RAILWAY ENGINEERING.

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DESIGNED FOR THE USE OF SCHOOLS AND PRIVATE STUDENTS.

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LONDON:

SPOTTISWOODE and SHAW, New-street-Square.

ΤΟ

J. P. KAY SHUTTLEWORTH, ESQ.

AND

E. C. TUFNEL, ESQ.

FOUNDERS OF THE BATTERSEA TRAINING COLLEGE,

This Work

IS

MOST RESPECTFULLY INSCRIBED,

AS

A TRIBUTE OF ESTEEM,

FOR THE IMPROVEMENTS THEY HAVE EFFECTED IN

OUR METHODS OF EDUCATION,

BY

THEIR HUMBLE AND MOST OBEDIENT SERVANT,

THE AUTHOR.

PREFACE.

THE present work is intended to supply Teachers and Practical Men with a simple and concise, yet comprehensive, system of theoretical and practical mathematics; embracing Geometry, Trigonometry, Mensuration, Surveying, and Levelling.

race.

In a course of elementary instruction, Geometry is, perhaps, not less important than Algebra. It will, therefore, be instructive to trace the origin of our ideas in geometry, with the view of suggesting to us the means whereby first notions, on this subject, should be conveyed to the mind of the learner. Geometry, no doubt, claimed attention at a very early epoch in the history of the human Surrounded by objects of geometrical symmetry,with all that is beautiful in form, in the earth, the sea, and the heavens, -man could not fail, even in the dawn of his intelligence, to trace the relations and properties of lines, surfaces, and solids. In the first rude essays at investigation, the scale and compasses would be used to confirm the deductions of reason, or the fortunate conjectures of intelligence; and a vast amount of geometrical facts would be accumulated, independently of the formality of definitions, or the tedious verbiage of a rigorous demonstration. The manner in which a figure is described, would invariably be taken as its definition; and thus no misunderstanding would take place with respect to the particular form considered, inasmuch as our conceptions arising from the perception of things, are always more vivid and positive, than those which are formed by mere verbal description. Results would be esteemed more valuable, than the processes by which they

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