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their reasons, as Seneca testifies of them, are not brought to persuade, but to compel.”

And in his inaugural speech before the University on his election to the Lucasian Professorship, he thus addressed the younger members of the University :

“But if nothing else can compel you to apply yourselves vigorously to these disciplines, and labour diligently in this exercise, at least that spur of noble minds, the thirst of following all great examples, which always sinks deep into generous dispositions, should the more strongly provoke and inflame you. Especially since you cannot merit the name nor sustain the dignity of an university upon any other account, than by preserving to yourselves a knowledge more than common in all kinds of science becoming a generous mind; and by taking away every occasion of calumny from all who either envy your happiness or emulate your fame. Not spending your time like infants in learning the languages, nor neglecting the care of searching after truth for digging up foolish fables out of the rubbish of obsolete antiquity; not affecting the vain trappings of words, and the delusions of a painted speech, while the nature of things lies unregarded, and the use of plain reason is set aside; nor, lastly, abusing your leisure, playing away your time, and miserably wresting and torturing your wits with sophistical trifles and empty jargon, engaging yourselves in barren disputes, insisting upon uncertain conjectures, and venting of doubtful opinions.

“These reproaches you may easily wipe off, or entirely avoid, by only applying yourselves to the study of the mathematics with that diligence which is requisite. The mathematics, I say, which effectually exercises, not vainly deludes nor vexatiously torments studious minds with obscure subtleties, perplexed difficulties, or contentious disquisitions; which overcomes without opposition, triumphs without pomp, compels without force, and rules absolutely without the loss of liberty ; which does not privately overreach a weak faith, but openly assaults an armed reason, obtains a total victory, and puts on inevitable chains; whose words are so many oracles, and works as many miracles; which blabs out nothing rashly, nor designs anything from the purpose, but plainly demonstrates and readily performs all things within its verge; which obtrudes no false shadows of science, but the very science itself, the mind firmly adhering to it as soon as possessed of it, and can never after desert it of its own accord, or be deprived of it by any force of others. Lastly, the mathematics which depend upon principles clear to the mind, and agreeable to experience ; which draws certain conclusions, instructs by profitable rules, unfolds pleasant questions, and produces wonderful effects; which is the fruitful parent of, I had almost said, all arts, the unshaken foundation of sciences, and the plentiful fountain of advantage to buman affairs."

ELEMENTARY ALGEBRA, WITH BRIEF NOTICES OF ITS HISTORY.

SECTION IV.

ADDITION AND SUBTRACTION,
MULTIPLICATION AND DIVISION.

BY ROBERT POTTS, M.A.,

TRINITY COLLEGE, CAMBRIDGE,
EON. LL.D. WILLIAM AND MARY COLLEGE, VA., U.S.

LONDON:
LONGMANS AND CO.,

The

versant with all subiossidered as speculativ

or decrease.

ALGEBRA.

EXPLANATIONS AND DEFINITIONS. Art. 1. As all our knowledge of the external world is subject more or less to the conditions of number and space, the sciences which treat of those subjects are of high importance.

The mathematical sciences may be considered as speculative or practical, and are conversant with all subjects which admit of increase or decrease. The speculative view regards the truth of the propositions, and the practical, the application of them to purposes of utility. The truth of the propositions depends on the definitions laid down as being intelligible, and the postulates and axioms assumed, as unquestionable. In the pure sciences of space and number no first principles are admitted, but such as are supported by their own strength and clear in their own light.

Arithmetic consists of methods for the performance of numerical calculations; Algebra is a generalisation and extension of Arithmetic.

The principles of Elementary Algebra have their origin in those of Elementary Arithmetic. In the latter, the symbols employed have each an intrinsic and local value, and the results of computation are definite; but in the former, general symbols are employed to denote quantities, unrestricted or limited in value; and operations and results only are indicated and not performed as in Arithmetic, but which may be performed in any case where numerical values are given to the general symbols.

The word quantity is assumed to mean both whatever can be made a subject of mathematical reasoning, and also the symbol whieh represents it, whether it be concrete or abstract. Two classes of symbols are employed, which may be described as “symbols of quan. tity” and “symbols of operation.”

2. The letters of alphabets are assumed to represent numbers and the magnitudes of all quantities, whether concrete or abstract.

The early letters a, b, c, &c., are generally assumed to denote known quantities, and the final letters x, y, 2, &c., unknown quantities. Besides these assumptions, other conventions are frequently employed, either to facilitate the solution of a problem, or to simplify the process of an investigation.

When a series of quantities is required to be expressed, instead of denoting them by the successive letters of the alphabet, they can be more conveniently denoted by the same letter repeated, with marks attached to it; as, for instance, instead of writing a, b, c, d, &c., a, , a", a'", &c., may be written; or still more conveniently ao, a 1, A2, A3, &c.

The series of the natural numbers beginning from 0, as 1, 2, 3, 4, 5, 6 ..... ..... may be supposed to be continued to any

ive lettere letter filing a, b, eveniently

assignable number n, and the series may be further extended beyond that limit to any extent. The symbol oo is assumed to denote & number greater than can be assigned.

3. In the science of Arithmetic numbers are considered as simply absolute, whether abstract or concrete; whereas in the science of Algebra both numbers and the general symbols of quantity are considered to have, besides their assumed values, some specific relations of an opposite or contrary nature. This idea of contrariety was originally denoted by a mark written above the symbol, and signified that it was to be considered in some way, of a contrary nature to one which was not so marked.

The marks + and – prefixed* to numbers and symbols denoting magnitude, are assumed to indicate opposite qualities, so that the symbols so marked are no longer absolute quantities, but quantities having some specific relation of an opposite nature to others with which they may be connected.

This idea of contrariety or opposition will be different according to the nature of the quantities considered, whether they are concrete or abstract quantities.

If a denote the length of a straight line, and if ta be assumed to donote a line of a units drawn in any direction; then – a may denote a line of the same length drawn in a contrary direction.

If o denote the magnitude of an angle generated by the motion of a moveable straight line round a point in a fixed straight line; and if to denote an angle traced out by the moving line in one direction in a plane from the fixed line, then –o may denote an equal angle traced out by the lice moving in a contrary direction.

If f denote the intensity of any force, and if + f be assumed to indicate a force acting in any direction; then -f may represent an equivalent force acting in an opposite direction.

In local motion, progression may be called affirmative, and regression, negative; and they may be denoted in the same manner; as if +8 denote a distance moved over in one direction, -8 may denote the same distance moved over in the contrary direction.

Periods of time before and after any fixed epoch may be indicated in the same manner; thus, if ta be assumed to denote a number of years after any fixed epoch, -a may consistently imply a period of the same number of years before that epoch.

If t denote any degrees of temperature, measured from some fixed

• The symbol +, by writers on Algebra in the Latin language, was named plus, more, and when it was placed before any number or symbol of quantity, was translated “increased by ;" also the symbol – was called minus, less, and when prefixed to any quantity, signified " lessened by" or "diminished by."

The sign X was a substitute for “ductum in,” multiplied into;” and “ was assumed for applicalum ad, and explained to mean "divided by.

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