our Author had been early and intimately acquainted with; witness his Geometria Organica, printed in 1719, when he was not full twenty-one years of age, and which, though fo juvenile a work, gained him, at once, that diftinguished Rank among Mathematicians, which he thenceforth held with great luftre. Yet he frankly owns, he was led to many of the Propofitions in this Appendix, from a Theorem of Mr. COTES, communicated to him, without any demonftration, by the Reverend and Learned Dr. SMITH, Mafter of TrinityCollege, Cambridge. How he has profited of that Hint, the Learned will judge: Thus much we can venture to say, that he himself fet fome value upon this Performance; having, we are told, employed fome of the latest bours he could give to fuch Studies, in revifing it for the Prefs; to bequeath it as his laft Legacy to the Sciences and to the Publick.

The gentlemen to whom Mr. MACLAURIN left the care of his Papers, are MARTIN FOLKES, Efq; Prefident of the Royal Society; ANDREW MITCHEL, Efq; and the Reverend Mr.HILL, Chaplain to his Grace the Archbishop of Canterbury; with whom he had lived in a moft intimate friendship. And by their direction this Treatife is published.

CON

XII. Containing fome general Theorems for

the exterminating unknown quanti-

ties in given equations

XIII. Of Quadratic equations

XIV. Of Surds

Supplement to Chap. XIV.

17

24

34

42

54

61

68

81

85

94

127

PART

PART II.

11

Of the Genefis and Refolution of equa-

tions of all degrees; and of the different

Affections of the roots.

Chap. I. Of the Genefis and Refolution of equa-

tions in general; and the number

of roots an equation of any degree

may have

Pag. 131

II. Of the Signs and Coefficients of equa-

tions

III. Of the Transformation of equations,

and exterminating their intermedi-

ate terms

IV. Of finding the Roots of equations

when two or more of the roots are

equal to each other

V. Of the Limits of equations

VI. Of the Refolution of equations, all

whofe roots are commenfurate

VII. Of the Refolution of equations by

finding the equations of a lower de-

gree that are their divifors

Supplement to Chap. VII. Of the Reduction

of equations by furd divifors

VIII. Of the Refolution of equations by

Cardan's rule, and others of that

kind

IX. Of the Methods by which you may

approximate to the roots of nu-

139

148

162

170

186

197

213

223

meral

meral equations by their limits

Pag. 231

Chap. X. Of the Method of Series, by which

you may approximate to the roots

of literal equations

XI. Of the Rules for finding the number

of impoffible roots in an equa-

tion

XII. Containing a general Demonftration of

Sir Ifaac Newton's rule for finding

the fums of the powers of the roots

of an equation

244

275

286

Of the Application of Algebra and Geometry

to each other.

Chap. I. Of the Relation between the equations

of curve lines and the figure of

thofe curves, in general

II. Of the Conftruction of quadratic

297

equations, and of the Properties of

the lines of the fecond order

III. Of the Construction of cubic and bi-

quadratic equations

APPENDIX.

325

352

De Linearum Geometricarum Proprietatibus

generalibus.

A TREA

Α

TREATISE

OF

ALGEBRA.

PART I.

CHAP. I. ·

Definitions and Illuftrations.

A

LGEBRA is a general method of computation by certain figns and fymbols which have been contrived for this purpose, and

found convenient. It is called an UNIVERSAL ARITHMETICK, and proceeds by operations and rules fimilar to those in common arithmetick, founded upon the fame principles. This, however, is no argument against its usefulness or evidence; fince arithmetick is not to be the lefs