OF

PLANE GEOMETRY

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.

BY NATHAN SCHOLFIELD.

BIBLI

LONDON:

WILEY AND PUTNAM, 6, WATERLOO PLACE.

PREFACE.

IN 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

after all, none have been able to supersede his authority in point of rigorous accuracy.

Among the many modern works on elementary Geometry may be mentioned the names of Lacroix and Legendre of the French, Hutton, Leslie and Young, of the English. Others of considerable merit might be mentioned, but these are specified on account of their popularity; Legendre particularly having given a more favorable impetus to speculative, and Hutton to practical Geometry, or Mensuration. By means of which, the science has been clothed in a more attractive form and elevated to a more favorable position. The elementary part of the work now presented to the public, the author has deemed it expedient to base on the elements of Legendre, not only on account of the elegance and general accuracy of that work, but also on account of its well deserved popularity. But while this work has been selected as the basis, others have been consulted, and whenever it was thought the subject required any change, selections have been made from other authors, under such modifications as were required to adapt them to this work, or the propositions in the original of Legendre have occasionally been slightly altered when propriety seemed to require it, and many original propositions have been added, and the work greatly extended both in plane and solid Geometry, but more particularly in solid.

It is needless here to specify all the supposed improvements introduced in the elementary parts of the work, but it is believed that some new definitions, which are here introduced, will be regarded with favor, as being more strictly in accordance with mathematical reasoning. The subject of Proportion, which is wanting in Legendre, and which is made the Fifth Book of Euclid's Elements, is here made to precede the other parts, thereby allowing those parts more purely geometrical to follow in consecutive order. The subject of Proportion is prefaced by the application of numbers to magnitudes, and the whole subject of proportional magnitudes is treated in a full, and, as the author conceives, in a perspicuous manner. The subject of incommensurable quantities or magnitudes is discussed, and it is shown that our reasoning in reference to such quantities ought not to be embarrassed on account of their incommensurability, since any magnitude may, as will be shown, be expressed by some function of any other magnitude, aud hence may have some definite expression in terms of such magnitude. A new mode of treating the subject of parallels, it is thought, pos