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AND

CYCLOPEDIA

OF

MATHEMATICAL SCIENCE.

COMPRISING

DEFINITIONS OF ALL THE TERMS EMPLOYED IN MATHEMATICS-AN
ANALYSIS OF EACH BRANCH, AND OF THE WHOLE,

AS FORMING A SINGLE SCIENCE.

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ASSISTANT PROFESSOR OF MATHEMATICS, UNITED STATES MILITARY ACADEMY.

NEW YORK:
PUBLISHED BY A. S. BARNES & Co.,

No. 51 JOHN-STREET.

1855.

HARVARD CI!!FREIGHARY

GIFT OF THE

EducT108.55.317

CONGREGATIONAL LIBRARY OF BOSTON

JAN 4 1937

DAVIES' COURSE OF MATHEMATICS.

Primary Arithmetic and Table Book-An entire new book, designed to take the place of " Davies' First Lessons.” It is composed of easy and progressive lessons, and adapted to the capacities of young children.

Intellectual Arithmetic, or AN ANALYSIS OF THE SCIENCE OF Numbers.—This is also a new book, and designed as a full and complete class-book for the advanced student of Mental Arithmetic in all our Public Schools and Academies. Great care has been taken in the arrangement and gradation of the lessons, in the character of the questions, and in the full, clear, and logical forms of the analysis.

New School Arithmetic, ANALYTICAL AND PRACTICAL, is a complete and thorough revision of the previous editions of his School Arithmetic. Much new matter has been introduced ; the arrangement is more natural and scientific ; the methods introduced are those used by some of the best teachers in the country.

University Arithmetic.—The object of this work is to give a general view of the Science of Numbers, and to point out all the general methods of their application.

Practical Mathematics for Practical Men.—The design of this work is to afford to the Schools and Academies an Elementary work of a practical character.

Elementary Algebra.- This work is intended to form a connecting link between Arithmetic and Algebra, and to unite and blend, as far as possible, the reasoning in numbers with the more abstract method of analysis. It is intended to bring the subject of Algebra within the range of our common-schools, by giving to it a practical and tangible form.

Elementary Geometry and Trigonometry.-This work is designed for those whose education extends beyond the acquisition of facts and practical knowledge, but who have not time to go through a full course of mathematical studies. It is intended to present the striking and important truths of Geometry in a form more simple and concise than is adopted in Legendre, and yet preserve the exactness of rigorous reasoning.

Elements of Surveying.-In this work, it was the intention of the author to begin with the very elements of the subject, and to combine those elements in the simplest manner, so as to render the higher branches of Plane Surveying comparatively easy. All the instruments needed for plotting have been carefully described, and the uses of those required for the measurement of angles are fully explained.

Bourdon's Algebra-New AND ENLARGED EDITION.—The Treatise on Algebra by M. Bourdon, is a work of singular excellence and merit. In France it is one of the leading textbooks. Shortly after its first publication it passed through several editions, and has formed the basis of every subsequent work on the subject of Algebra

Legendre's Geometry and Trigonometry–ReviSED Edition.-Legendre's Geometry has taken the place of Euclid, to a great extent, both in Europe and in this country.

Analytical Geometry.-This work embraces the investigation of the properties of geometrical figures by means of analysis.

Descriptive Geometry.-Descriptive Geometry is intimately connected with Architecture and Civil Engineering, and affords great facilities in all the operations of Construction.

Shades, Shadows, and Perspective. This work embraces the various applications of Descriptive Geometrý to Drawing and Linear Perspective.

Differential and Integral Calculus.- This Treatise on the Differential and Integral Calculus, is intended to supply the higher seminaries of learning with a text-book on that branch of science.

Logic and Utility of Mathematics is an elaborate exposition of the principles which lie at the foundation of pure mathematics, and of the applications of those principles to the development of the essential idea of Arithmetic, Geometry, Algebra, Analytic Geometry, and the Differential and Integral Calculus.

Mathematical Dictionary and Cyclopedia of Mathematical Science-Embracing the definitions of all the terms of Mathematical science, an analysis of each branch, and of the whole, as forming a single science-designed especially to illustrate the entire course.

Entered, according to act of Congress, in the year Eighteen Hundred and Fifty-five, by CHARLES DAVIES

& WILLIAM G. Peck, in the Clerk's Office of the District Court of the United States, for the Southern

District of New York.
JONES & DENYSE, STREXOTYPERS.

G. W. WOOD, PAINTER,

PREFACE.

The Science of Mathematics treats of the two abstract quantities, Number and Space. Primarily, it treats of the measurements and relations of these quantities, and of the operations and processes by means of which they are ascertained : and secondarily, of the applications of the principles thus developed to the practical affairs of life.

The quantities operated upon are denoted by figures or letters, and the operations to be performed are indicated by certain characters called Signs. The figures, letters and signs, are called symbols, and are elements of the mathematical language.

The language of mathematics is partly technical and partly popular, being made up of symbols which either represent quantity or denote operations, and of words adopted from our common vocabulary. Both branches of this language are undergoing changes corresponding to the progress and development of the science ; and hence it is, that new terms become necessary, while the significations of the old ones are modified, either by enlargement or restriction.

It is of the first importance, in prosecuting mathematical inquiries, to acquire an accurate knowledge of the office and power of every symbol, and a clear and distinct apprehension of the signification of every technical term. Most of the difficulty experienced in the study of mathematics, has arisen, we apprehend, from the use of terms in a vague or ambiguous sense ; and the discussions on “controverted points,” are mainly due to a misuse or misapprehension of the meaning of technical terms.

1. It is a leading object of this work, to define, with precision and accuracy, every term which is used in mathematical science; and to afford, as far as possible, a definite, perspicuous and uniform language.

2. A second object is, to present in a popular and condensed form, a separate and yet connected view of all the branches of Mathematical Science. Hence, the work has been called "A DICTIONARY AND CYCLOPEDIA OF MATHEMATICAL SCIENCE."

3. The work has also been prepared to meet the wants of the general reader, who will find in it all that he needs on the subject of mathematics. He can learn from it the signification and use of every technical term, and can trace such term, in all its connections, through the entire science. He will find each subject as fully treated as the limits of the work will permit, and the relations of all the parts to each other carefully pointed out.

4. The practical man will find it a useful compendium and hand-book of reference. All the formulas and practical rules have been collected and arranged under their appropriate heads.

5. The chief design of the work, however, is to aid the teacher and student of mathematical science, by furnishing full and accurate definitions of all the terms, a popular treatise on each branch, and a general view of the whole subject.

In pursuing a course of mathematics, arranged in a series of Text-Books, it is often difficult, if not impossible, to understand a single branch fully until its connections with other branches shall have been traced out. The various branches of mathematics, though apparently differing widely from each other, are, nevertheless, pervaded by common principles and connected by common laws. In bringing all these branches within the compass of a single volume, an opportunity has been afforded of examining their common principles and pointing out the connections of their several parts. Hence, the Dictionary affords to the diligent and intelligent student, the means of understanding the connections of the different subjects of the mathematical science; and to such, we are confident, it will prove an efficient auxiliary in removing the obstacles which have rendered the acquisition of mathematical science a difficult and forbidding task.

The diffusion of knowledge and the employment of mathematics in the investigations of the Natural Sciences, as well as in all practical matters, have given great value to mathematical acquirement, if they have not rendered a certain amount of it absolutely necessary; hence, it would seem desirable to afford every facility for the prosecution of so useful a study.

As many of the subjects treated in this work have common parts, it became necessary either to interrupt the processes of investigation by references, or to use, occasionally, the same matter in different places. As the entire work is rather a collection of separate treatises than a single treatise on a single subject, the latter method has occasionally been adopted, though the other has been generally used.

It will not be a matter of surprise, that a work of so much labor should have been a joint production. In its prosecution, many questions have arisen in regard to definitions, methods of discussion, classification and arrangement. In deciding these points we have been guided, uniformly, by the best standards. When differences were irreconcilable we have looked to the authority of general principles.

FISHKILL LANDING, L

June, 1855.

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