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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.







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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.

In Algebra, it is employed to denote a known or given quantity-In Geometry and Trigonometry it often stands for an angle-5 In Surreying 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 cal- The vertical bar EF divides the rectangle culator.

AC into two separate compartments, but is AB'A-CUS. (L. abacus, anything flat. Gr. placed far enough in front of the wires to apaz, a slab for reckoning on). In architec- allow the beads or counters to slide freely ture, a table constituting the crowning mem- behind it. ber of a column and its capital.

We have supposed the instrument arranged The name abacus was given to an instru-according to the decimal scale, so that each inent formerly used to facilitate arithmetical bead on the lower wire denotes a simple unit, computations, and still retained in one of its or unit of the first order; a bead upon the modifications, for imparting to children a second wire, a unit of the second order, or knowledge of the elements of arithmctic. one ten; one upon the third wire, a unit of

The most ancient abacus consisted of a the third order, or one hundred; &c. table, surrounded by a raised border or ledge, Sometimes, for the purpose of diminishing and covered with sand, upon which diagrams the number of counters, intermediate wires were drawn, and computations made. 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.

Let the number in question be 7,931,564 The principle of this kind of abacus will First slide four beads along the lower wire be readily understood by an examination of into the compartment AF, these will denote the diagram.

the four units; then slide six bearls along the


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second wire to denote the six tens, and so on,

4a2 /3 - 2a$ /?, moving into the space AF a sufficient number may be abbreviated to the expression, of counters on each wire to denote the num

2a2 b(26 a), ber of units of the corresponding order ; when the operation

complete, the given by simply factoring it. number will be represented as in the diagram.

An abbreviation is a single letter, or a We have supposed the value of the unit, in simple combination of letters, standing for a passing from wire to wire, to increase ac- word or sentence: thus, A. stands for acre, cording to the decimal scale; but the instru- hhd. for hogshead, lb. for pound, &c. ment is equally applicable when the value of A-BRIDGE', [Fr. abréger, to shorten. Gr. the unit increases according to any scale, Bpaxus, short]. To shorten, to contract. The either uniform or varying. If the duodeci- abridgment of an expression in Algebra, is mal scale be adopted, there will be required the operation of shortening it by substitution : eleven counters on each wire ; if any varying thus, every equation of the second degree scale is used, the number of counters on any containing but one unknown quantity, is a wire must be at least equal to the number of particular case of the general form units of that order contained in a unit of the

aro + bx + c = 0; next superior order, diminished by 1. The

or, dividing both members by a, method of recording a number constructed ac

b cording to any scale, is entirely similar to

za + x + that already explained.

During the middle ages, the abacus was which may be abridged by substituting 2 p used by bankers, money-changers, &c., but b instead of a frame with wires and beads, they

for and a for giving the equation, made use of a bench or bank covered with black cloth, divided into checks by white lines

2a + 2 px + 9 = 0. at right angles to each other. Counters The last equation is not only easier to replaced upon the lower bar denoted pence, member, but is also under a simpler form for those on the second bar shillings, those on discussion. the third, pounds, and those on the fourth, The operation of abridging may generally fifth, sixth, &c., bars, denoted tens, hundreds, be resorted to with advantage, whenever comthousands, &c., of pounds.

plicated expressions enter into long compuAbacus Pythagoricus, or Pythagoreau tations. After completing the computations, abacus. A table computed for the purpose of we can, if necessary. substitute for the symfacilitating numerical calculations. It is bols introduced, their values in terms of the nothing more than the multiplication table as original quantities employed. given in ordinary treatises on arithmetic. AB-RUPT POINT of a curve. A point

Abscus Logisticus, or sexagesimal canon. at which a branch of a curve terminates : The same as the Pythagorean abacus or mul- thus, the curve whose equation is tiplication table, carried to 60, both ways.

y - b = (1 a)l(x a) AB-BRE’VI-ĀTE. [L. abbrevio, to shorten). has an abrupt point for x = a. See SinguTo reduce to a small compass, to epitomize.

AB-BRE-VI-Ā'TION, the operation of ab- AB-SCIS'SA. (L. abscissus, ab, from, and breviating or shortening. The abbreviation scindo, to cut]. One of the elements of refof a fraction is the operation of reducing it erence by means of which a point is referred to lower terms ; thus, if both numerator and to a system of rectilineal co-ordinate axes. denominator of the fraction be divided by If we draw two straight lines, in a plane, in9, the fraction is said to be abbreviated, or re- tersecting each other, one of them being duced to . In Algebra, an expression is horizontal, it has been agreed to call the said to be abbreviated when it is shortened horizontal one the axis of X, or the axis of by any algebraic process : thus, the expres- abscissas, and the other one the axis of Y, 0 sion

the axis of ordinates.

lar point.


In such a system, the abscissa of a point | magnitude represented by the equation passes is the distance cut off from the axis of X by through the origin of co-ordinates. See Ana line drawn through it, and parallel to the alytical Geometry. axis of Y. All abscissas measured to the

ABSOLUTE Space, is space considered withright, are, by convention, regarded as positive, out reference to material objects, or limits. and consequently, all at the left must be con

ABSTRACT. [L. abstractus, to draw from sidered negative. The abscissas of all points situated on the axis of Y, are 0. In space, draw). Separate, distinct from something else.

or separate ; from abs, from, and traho, to the term abscissa is applied in a more general sense, and may mean a distance meas

ABSTRACT EQUATION, is an equation exured parallel to either of the horizontal axes, pressing a relation between abstract quantithe distance measured on a parallel to the ties only, as, axis of Z being always called the ordinate.

3x2 + 4x - 5 = 0. It is customary to define the abscissa of a

ABSTRACT QUANTITY, is one which does not point in space, to be the distance of the point involve the idea of matter, but simply that of from the co-ordinate plane YZ, measured on a mental conception ; it is expressed by a a line parallel to the axis of X. The rule letter, symbol, or figures : thus, the number for signs is analogous to that employed in a three represents an abstract idea, that is, ono plane system ; all distances measured towards which has no connection with ma&rial things, the right are considered positive, those to the whilst three feet, presents to the mind an idea left must be negative; the abscissas of of a physical unit of measure, called a foot. points in the plane Y Z are 0. When the So a “portion of space bounded by a surface, term abscissa is applied to distances measured every point of which is equally distant from from the plane XZ and parallel to the axis of a point within, called the centre," is a mere Y, they are considered positive when meas- conception of form. When we call it a ured in front of the plane, and negative when sphere, we employ the term to express our measured behind it. When the point lies in idea of the abstract magnitude. the plane XZ, the abscissa is 0.

All numbers are abstract when the unit is AB'SO-LUTE, [L. absolutus, ab, from, and abstract. Arithmetic, which treats of the relasolto, to loose or release]. Complete in itself, tions and properties of such numbers, is abindependent. The absolute term of an equa

stract arithmctic. This embraces the whole tion, is that term which is known, or which science and theory of arithmetic: Concrete or does not contain the unknown quantity: thus, Denominate Arithmetic being nothing more in the equation

than the art of applying the principles devel

oped in Abstract Arithmetic to Denominato al' + bra + cx + d=0,

Numbers. d is the absolute term. If we regard every

Since Algebra differs little from ordinary term as involving the unknown quantity in Arithmetic, except in the nature of the lan. some form, then the absolute tern is that in guage employed, we must regard the Scienco which the exponent of this quantity is 0. In of Algebra as purely abstract. In Geometry every entire equation, the absolute term is also, the magnitudes considered, viz., lines equal to the continued product of all the roots surfaces and solids, are mere mental concepof the equation with their signs changed. tions of extent and form, which are repreHence, if the absolute term of an equation sented by geometrical figures. The discussion is 0, one or more of the roots of the equation of these magnitudes in the development of must be equal to 0.

their relations and properties is, therefore, In Analytical Geometry, the equations em- necessarily confined to abstract quantities. ployed are indeterminate; that is, they involve We may, therefore, regard Geometry as an more unknown quantities than there are equa- Abstract Science. And generally, all the tions, and the absolute term is that one which principles of Mathematical Science are deis independent of all the unknown quantities veloped from a consideration of abstract quan or variables. It may be demonstrated that tities only. when the absolute term is 0, the geometrical What is usually termed Abstract Mathe

19 271


matics, or pure mathematics, involves the tra-distinction to that of descending ground, entire science, whilst that which is called Con- which is called declivity. Acclivity implies crete, or Mixed Mathematics, is nothing more ascent, declivity implies descent. than the art of applying previously developed

A'CRE (L. ager, land. Gr. aypos, a field). principles to physical objects, as suggested A unit of measure employed in land survey. by the demands of society.

ing. In the United States, the standard acre AB-SURD'. [L. absurdus, ab, from, and sur- contains 4840 square yards, or 43,560 square dus, deaf, insensible). A proposition is absurd, feet. In the form of a square, one side would when it is opposed to a known truth. The measure about 69.5701 yards, or 208.7103 feet. term absurd, is used in connection with a

The acre contains 160 square rods, or perkind of demonstration called the reductio ad ches. The subdivisions of the acre are roods absurdum.” In this kind of demonstration, and perches. The acre containing 4 roods, a certain proposition is assumed as true, and and the rood 40 perches. is combined with known truths by a course

There are 640 acres in a square mile, hence, of logical arguments, thus deducing a chain an acre is the nigth part of a square mile. of conclusions until one is arrived at which

The English statute acre is the same as disagrees with a known truth, when the origi- that of the United States. nal supposition or hypothesis is pronounced

The Irish acre contains 1 acre 2 roods absurd, and its contrary is considered proved.

perches English. As an example of this mode of demonstra

The Scotch acre contains 1 acre 1 rood tion, we may instance the proposition to show

871 perches English. that, “If two straight lines have two points

The Welsh acre contains about 2 English in common,

they will coincide throughout." In this proposition it follows that if they do

The Strasbourg acre is about one-half of not coincide, they must enclose a space which

an English acre. is manifestly impossible ; hence, the proposition that they do not coincide involves an tains 4,088 square yards, or nearly ☆ of an

The French acre or arpent of Paris conabsurdity, and the proposition is said to be re

English acre. duced to an absurdity. This method of proof,

The French woodland arpent contains 6,108 though sometimes objected to as unsatisfactory, is, nevertheless, as strictly logical, and square yards, or about 1 acre 1 rood 1 perch

English as conclusive as any other method. The rea

In the new decimal system of France, the soning is quite as perfect, and the conclusions

Are contains 119.603 square yards, the Decequally irresistible.

are 1196.03 square yards, and the Hecatare A-BUNDANT NUM'BER. A number

11960.3 square yards. which is less than the sum of all its aliquot

A-CUTE', [L. parts : thus, 12 is an abundant number, be

cutus, sharp pointed]

Sharp as opposed to obtuse. An acute angle, 12 <1+2 +3 +4 + 6.

is one that is less than a right angle. In deAn abundant number is distinguished from a grees, an acute angle is less than 90°. perfect number, which is equal to the sum of ACUTE-ANGLED TRIANGLE, is one that has its aliquot parts, and from a deficient number, all of its angles acute. which is greater than the sum of its aliquot ACUTE CONE, is one in which the vertical parts.

angle of the meridian triangle is less than AC-CI-DENT'AL POINT of a line. In 90°, or less than a right angle. Perspective, the point in which a line drawn

ACUTE HYPERBOLA, is one whose asympthrough the point of sight, and parallel to the

totes make an acute angle with each other. given line, pierces the perspective plane. It is a point of the indefinite perspective of the In it, the transverse axis is always greater

than the conjugate. line. See Vanishing Point.

AC-CLIV'I-TY, [L. acclivus, from ad, to, ADD, (L. addo, from ad, to, and do, to and clivus, an ascent). In Topography, the give]. To unite or put together, so as to steepness or slope of rising ground, in con- form an aggregate of several particulars.


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