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nitude, and figure; it becomes accustomed to the contemplation of forms, and acquires a certainty and readiness of the imagination which enables it to make with variety and skill, new combinations of the elements of form. Nor is Descriptive Geometry confined to the mere representation of visible forms; many of the severer subjects belonging to mathematical science, are thoroughly discussed by processes which this science has taught. “In a word, with the aid of a small number of the elementary propositions of geometry, this science possesses an almost infinite variety of means, by which we may arrive at the solution of the most difficult problems. It requires but a few weeks' study to be sufficiently understood; it advantageously replaces the common modes of practice, the long and laborious study of which is rendered unnecessary ; at the same time, it gives us the immense advantage of treating with equal ease new combinations or unforseen cases.””

In offering to the public this Introduction to Descriptive Geometry, the Author makes no pretensions to originality. His object has been to comprise, within convenient limits, the fundamental principles of the science first given to the world by Monge, with a few examples of its application to linear perspective and some other important projections. It is, however, but an introduction; and though it gives the general principles of construction, is not meant to be considered as a treatise. If it should serve the purpose of calling the attention of teachers and the guardians of our academical institutions, to the importance of the subject, the object of this part of the work will have been obtained.

In preparing the Introduction to Descriptive Geometry, the writings of Monge, Hachette, and Lacroix have been

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consulted, and also the treatises upon this subject of Cloquet and Professor Davies of West Point. To the latter treatise the reader is referred for a fuller discussion of Tangent and Cutting Planes and Spherical Projections; also for a discussion of Spherical Trigonometry and the subject of Warped Surfaces. In Cloquet's treatise Descriptive Geometry is applied to the projection of a great variety of bodies, to the laws of Optics, to the determination of shadows, and to the science of perspective drawing. In the present treatise, the chapter upon Perspective is mostly taken from Lacroix.

There will be found, both in the Elements and in the Introduction to Descriptive Geometry, errors and slight omissions which can hardly be avoided in a first impression of a work of this kind; but none, it is hoped, which essentially affect the work.

Cambridge, October 28, 1829.

ELEMENTS OF GEOMETRY.

GENERAL NoTIONs of ExTENSION.

1. GEOMETRY is that science which teachès us to investigate the magnitudes and forms of extended things, or cztended space, and the relations of their parts.

2. A body, or the space which it occupies, is extended in three directions ; it has length, breadth, and thickness or depth. If it were destitute of either of these dimensions, it would cease to be. , 3. The space which a body occupies, is separated from

other space by what we call the surface or the outside of

the body. A surface is extended in two directions ; it has length and breadth, but is destitute of thickness, and therefore makes no part of the body itself. A body then, and even definite space, is bounded by surface.

4. If the body have several faces, like a square block of wood, for instance, these faces may be considered as so many distinct surfaces, each of which is bounded by the edges formed by the meeting of this face with the other faces of the body. These limits are no part of the surface ; they have neither breadth nor thickness; they have only length, and are called lines. The limit of a surface, therefore, is a line.

5. The line itself is limited by a point; which has no extension. A point may also be taken in a line, a surface, a body, or in extended space; it, however, makes no part of either of these magnitudes. It has position, but no extent. It may, by moving in space, be considered as generating a line.

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