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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.

BY

CHARLES DAVIES, LL. D.,

AUTHOR OF A COMPLETE COURSE OF MATHEMATICS,

AND

WILLIAM G. PECK, A. M.,

ASSISTANT PROFESSOR OF MATHEMATICS, UNITED STATES MILITARY ACADEMY.

A. S. BARNES AND COMPANY,
NEW YORK AND CHICAGO.

1872

KF 3860

DAVIES' MATHEMATICS.

THE WEST POINT COURSE,
And Only Thorough and Complete Mathematical Series.

IN THREE PARTS.

I. COMMON SCHOOL COURSE.

Davies' Primary Arithmetic.-The fundamental principles displayed in
Object Lessons.

Davies' Intellectual Arithmetic.-Referring all operations to the unit 1 as
the only tangible basis for logical development.

Davies' Elements of Written Arithmetic.-A practical introduction to the whole subject. Theory subordinated to Practice.

Davies' Practical Arithmetic.*-The most successful combination of Theory and Practice, clear, exact, brief, and comprehensive.

II. ACADEMIC COURSE.

Davies' University Arithmetic.*-Treating the subject exhaustively as a science, in a logical series of connected propositions.

Davies' Elementary Algebra.*-A connecting link, conducting the pupil easily from arithmetical processes to abstract analysis.

Davies' University Algebra.*-For institutions desiring a more complete but not the fullest course in pure Algebra.

Davies' Practical Mathematics.-The science practically applied to the useful arts, as Drawing, Architecture, Surveying, Mechanics, etc.

Davies' Elementary Geometry. The important principles in simple form, but with all the exactness of vigorous reasoning.

Davies' Elements of Surveying.-Re-written in 1870. The simplest and most practical presentation for youths of 12 to 16.

III. COLLEGIATE COURSE.

Davies' Bourdon's Algebra.*-Embracing Sturm's Theorem, and a most exhaustive and scholarly course.

Davies' University Algebra.*-A shorter course than Bourdon, for Institutions have less time to give the subject.

Davies' Legendre's Geometry.-Acknowledged the only satisfactory treatise of its grade. 300,000 copies have been sold.

Davies' Analytical Geometry and Calculus.-The shorter treatises, combined in one volume, are more available for American courses of study. Davies' Analytical Geometry. The original compendiums, for those deDavies' Diff. & Int. Calculus. siring to give full time to each branch. Davies' Descriptive Geometry. With application to Spherical Trigonometry, Spherical Projections, and Warped Surfaces.

Davies' Shades, Shadows, and Perspective.-A succinct exposition of the mathematical principles involved.

Davies' Science of Mathematics.-For teachers, embracing

I. GRAMMAR OF ARITHMETIC,

II. OUTLINES OF MATHEMATICS,

III. LOGIC AND UTILITY OF MATHEMATICS,
IV. MATHEMATICAL DICTIONARY.

* Keys may be obtained from the Publishers by Teachers only.

Entered, according to Act of Congress, in the year 1855, by
CHARLES DAVIES & WM. G. PECK,

In the Clerk's Office of the District Court of the United States for the Southern District of

M. D.

New York.

HARVARD
UNIVERSITY

LIBRARY

PREFACE.

1HE 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 neces sary 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.

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MATHEMATICAL DICTIONARY

AND

CYCLOPEDIA OF MATHEMATICAL SCIENCE.

A. The first letter of the English alphabet. | ABCD is a wooden frame, supposed to be Among the ancients it was used as a numeral vertical, supporting several horizontal wires, denoting 500, or with a dash over it, thus, A. 1, 2, 3, 4, &c. ; each wire bearing nine beads it stood for 500,000. In Greek, Hebrew, and of glass, wood, or ivory, which slide freely. Arabic, it stood for 1.

7

6

In Algebra, it is employed to denote a known or given quantity-In Geometry and Trigonometry it often stands for an angle 5 In Surveying it is used as an abbreviation for acre-In Commerce it stands for accepted, as in the case of a bill of exchange.

AB'A-CIST, [from abacus]. One who makes arithmetical computations, a computor or calculator.

AB'A-CUS. [L. abacus, anything flat. Gr. aßaş, a slab for reckoning on]. In architecture, a table constituting the crowning member of a column and its capital.

The name abacus was given to an instrument formerly used to facilitate arithmetical computations, and still retained in one of its modifications, for imparting to children a knowledge of the elements of arithmetic.

The most ancient abacus consisted of a table, surrounded by a raised border or ledge, and covered with sand, upon which diagrams were drawn, and computations made.

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3

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millions.

hun, of thous

tens of thous thousands. hundre

tens.

units.

The vertical bar EF divides the rectangle AC into two separate compartments, but is placed far enough in front of the wires to allow the beads or counters to slide freely behind it.

We have supposed the instrument arranged according to the decimal scale, so that each bead on the lower wire denotes a simple unit, or unit of the first order; a bead upon the second wire, a unit of the second order, or one ten; one upon the third wire, a unit of the third order, or one hundred; &c.

Sometimes, for the purpose of diminishing the number of counters, intermediate wires are introduced; a bead upon one of them In later times the sanded table was re- denoting five beads upon the wire next below placed by an entirely different instrument, We shall now explain the mode of recording a called also an abacus, and which with little number by means of this abacus. The beads change of principle, continued to be used for are all pushed along the wires into the commany centuries, by the most enlightened na-partment EC before the record is commenced. tions of the world.

The principle of this kind of abacus will be readily understood by an examination of the diagram.

Let the number in question be 7.931,564. First slide four beads along the lower wire into the compartment AF, these will denote the four units; then slide six bearls along the

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