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Some intricacies belonging to equations of the third degree were further unravelled by Raphael Bombelli, of Bologna, and published in 1579.

The algebraists, whose labours have been noted, expressed known quantities by their proper numerical characters, and therefore, their solutions were destitute of that generality which constitutes so prominent a feature of our present analysis. Their modes of solution were applicable to all similar problems, but their results were confined to particular questions. Francis Vieta, who was born at Fontenoy in Poitou, in 1540, and died in 1603, by using letters of the alphabet to represent known as well as unknown quantities, gave an extent and generality to the science which it did not possess before. By this change, algebraists were enabled to include, in a single solution, a whole class of problems, and to obtain the result in each particular case, by simple substitution. He taught the method of taking, from an equation, its second term, and thus reducing adfected quadratics, to simple quadratics; and all cases of cubic equations to the case solved by Ferrei. He also taught the solution of cubic equations, having three possible roots, by the trisection of an angle. He made numerous improvements in algebra, and furnished the germs of some discoveries which have since grown up under other names.

Thomas Harriott, an English analyst, followed in the steps of Vieta, and made several important improvements in the science. He, first, adopted the plan of placing all the terms of an equation on one side of the sign of equality, and zero on the other; and showed that an equation thus expressed, may be always formed by the multiplication of binomial factors. This naturally led to the discovery, that every equation has as many roots, or values of the unknown quantity, as there are units in the index of its highest power. Harriott, notwithstanding this observation lay directly in his road, does not appear to have made it. The complete developement of negative and impossible roots, was left to exercise the ingenuity of succeeding inquirers. Vieta employed the large letters of the alphabet, and indicated

their powers by initials placed over them, as exponents are now used: Harriott substituted small letters, and denoted their powers by repetitions of the letter; thus instead of A, Ac, &c. he wrote aa, aaa, &c., a small change indeed, but still an obvious improvement in the notation.

His Artis Analytica Praxis, containing his discoveries, was published after his death.* He was born at Oxford in 1560, and died in 1621. The celebrated French philosopher, René Descartes, contributed largely to the advancement of algebra. He explained the nature of negative roots, and taught the manner of finding their number by the changes of the signs in the general equation. The use of exponents, as now applied, is attributed to him; as is also the method of indeterminate co-efficients.

The first application of algebra to geometry, was long prior to the time of Descartes, yet those sciences are indebted to him for that intimate union, which has since contributed so extensively to the improvement of both. Descartes was born in 1596, and died in 1650.

Soon after the time of Descartes, the analytical science took a flight, which if followed, would lead me far beyond the bounds of a preface. This sketch of the history will, therefore, be closed with the remark, that this science, as enriched by the discoveries of Newton, Leibnitz, and others, has become in the hands of our modern philosophers, the torch to guide them through the most intricate labyrinths of science; that by its light they have traced the motions of the celestial bodies through all their mystic dance, and penetrated many of the recesses of nature, where, without its aid, they must have been bewildered and lost.

The following work was undertaken from a persuasion, that the books on algebra, used in our schools, were none of them, entirely adapted to the wants of a large class of pupils, many of whom do not enjoy the leisure,

*

Bossut says it was published in 1620, Montucla and others say in 1631.

or the talents requisite for penetrating the depths of science, and yet are desirous of attaining a knowledge of this subject, sufficient to qualify them for studying successfully the common practical branches of mathematics, to which this serves as a key. In the most popular treatise on this science, with which our schools have been furnished, the progress of the student appears to me, needlessly obstructed, by difficulties near the commencement, which to a common intellect, are almost insuperable.

My object has been to present the most useful parts of the science in such order, that no very abstruse process should be required, before the pupil had been sufficiently exercised, to acquire the requisite skill. The expedients demanded for solving the questions, are mostly pointed out before they are called for in practice.

In a popular treatise on a subject which has engaged the attention of so great a number of authors, many of them, unquestionably, endued with talents of the highest order, it would be idle to expect much originality. My object has been to smooth the path of the student, and diminish the toil of the tutor. The work, with all its imperfections, is submitted to the inspection of the public.

THE

PRACTICAL ANALYST.

DEFINITIONS.

ARTICLE 1. Algebra, or specious arithmetic, is the science of computing by symbols or general characters.

2. Quantities, of whatever kind, are usually denoted by letters of the alphabet.

3. The relations of quantities, and the operations to be performed on them, are indicated by the following characters, thus:

4. The sign plus, or more, indicates addition, as a+b, signifies that b is added to a.

5. The symbol-minus, or less, indicates subtraction; thus, a-b, signifies that b is subtracted from a.

The characters + and are called, by way of eminence, the signs of the quantities to which they are prefixed.

6. Multiplication is denoted by the sign X, into, placed between the factors, as axb; or by a period, as a.b; or, more frequently, by joining the letters, like letters in a word, as ab; each of which expressions denotes the product of a and b.

7. Division is indicated by the sign÷by, placed between the terms; or by writing the dividend above, and the divisor below a horizontal line; thus ab, or de

notes the quotient of a when divided by b. B

α

b

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