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JULIUS PETERSEN,
PROFESSOR AT THE ROYAL POLYTECHNIC SCHOOL AT COPENHAGEN

AND MEMBER OF THE ROYAL DANISH SOCIETY OF SCIENCE.

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LONDON
SAMPSON LOW, MARSTON, SEARLE & RIVINGTON

COPENHAGEN: ANDR. FRED. HÖST & SON

1880

(All rights reserved)
163 6. 3.

Printed by Bianco Luno.

INTRODUCTION.

1. Every object which occurs in nature has numerous properties; to facilitate the investigation of which, one preliminarily endeavours to consider one at a time, independently of the others; the different investigations of the same kind are then collected, and thus the different sciences, natural sciences, are formed. Thus, if we examine a piece of chalk, we might ask about its origin and occurrence, its specific gravity and colour, the combination of its elements, its form, &c., and these questions will be answered respectively by Geology, Physics, Chemistry, and Geometry.

2. Geometry treats of the form without regard to the substance; when we speak of a sphere, we do not consider of what it is made, but only of the space which it occupies; every object occupies a space, which has extension in all directions; this is a geometrical body.

The boundary between a body and the surrounding space is called a superficies or the surface of the body; a superficies has no thickness.

If a part of a superficies be cut off from the surrounding part, the boundary is called a line; a line has no breadth. If a part of a line be cut off, the limit is termed a point; it has no magnitude, but only denotes position.

3. When a point moves, it produces a line, similarly a superficies may be imagined as being produced through the

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