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OF

PLANE GEOMETRY

AND

AND

MENSURATION,

ON THE BASIS OF LEGENDRE'S ELEMENTS, WITH NUMEROUS

ALTERATIONS AND ADDITIONS:

BEING THE

FIRST PART OF A SERIES

ON

ELEMENTARY AND HIGHER

GEOMETRY, TRIGONOMETRY, AND MENSURATION;

CONTAINING MANY VALUABLE DISCOVERIES AND IMPROVEMENTS IN MATHEMATICAL
SCIENCE, ESPECIALLY IN RELATION TO THE QUADRATURE OF THE CIRCLE,
AND SOME OTHER CURVES, AS WELL AS THE CUBATURE OF CERTAIN
CURVILINEAR SOLIDS; DESIGNED AS A TEXT-BOOK FOR COLLEGIATE
AND ACADEMIC INSTRUCTION, AND AS A PRACTICAL

COMPENDIUM ON MENSURATION.

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LONLON:
WILEY AND PUTNAM, 6, WATERLOO PLACE

PREFACE.

Is presenting this work to the public, the author is fully impressed with the responsibility he assumes, since the press is constantly teeming with works of a similar character, from the pens of the most eminent mathematicians in this and other countries; and since we have the accumulated discoveries of ancient and modern times, the condensed wisdom of ages, the result of talent and labor, spread out on pages before us, it may by some be supposed that no new truths, no new principles or applications can be elicited, especially in elementary Geometry, which reached almost its present perfection in a remote period of antiquity. The works of Euclid, for more than twenty centuries, having been and are still, with scarcely an important improvement, the standard works on elementary Geometry; and while other sciences have been progressing and vieing with each other in the splendor of progressive acquisition, this has been handed down to us as a system of unchangeable truths, not affected by time or varying circumstances; evidently showing the science to be based on immutable principles. But although the elements of this science were thus early established, yet by the application of the principles therein contained, new truths have been constantly developing themselves in an accelerated progression, and many important mathematical truths which have been supposed unattainable by human ken have thereby been established on the same basis as the elements themselves.

It may be possible, therefore, that other mathematical principles, either elementary or general, may yet be developed by the skilful application of those already established.

It is with this view that the author has undertaken this work; having, as he deems, made some advancements in the science, not only in the elements, but also in its general application.

Although Euclid's elements deservedly possess such high regard or merit, both on account of their antiquity and the rigorous accuracy of his reasoning, yet we have numerous modern works on the subject, some of which are better adapted to the present state of science, the authors of which only aimed at the perfection of Euclid, with a more obvious connection of the parts, or greater elegance of diction; but

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