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fionally improving to the Perfection be intended if should bave, before be.committed it to the Press. And the beft Copies of his Manuscript having been transmitted to the Publifher, it was easy, by comparing them, to establish a correct and genuine Text. There were, besides, several detached Papers, some of which were quite finished, and wanted only to be inserted in their proper places. In a few others, the Demonstrations were so concisely expressed, and couched in Algebraical characters, that it was necessary to write them out at more length, to make them of a piece with the rest. And this is the only liberty the Publisher bas allowed bimself to take ; excepting a few inconsiderable additions, that seemed necefsary to render the Book more complete within itself, and to save the trauble of consulting others who have written on the fame Subject.

The Rules concerning the Impoflible Roots of Equations, our Author had very fully considered, as appears from his Manuscript Papers: but as he had no where reduced any thing on that Subje&t to a bétter form, than what was long ago published in the Philosophical Transactions, No. 394 and 408, we thought it best to take the substance of Chap. 11. Part II. from thence; especially af the latter of these Papers furnishes a demonftration

of the original Rule, which pre-fupposes only what the Reader has been taught in a preceding Chapter.

The Paper that is subjoined, on the Sums of the Powers of the Roots of an Equation, is taken from a Letter of the Author (8 Jul. 1743) to the Right Honourable the Earl STANHOPE; communicated to the Publisher, with some things added by his Lordship, which were wanting to finish the Demonstration.

Of these Materials, carefully collected and put in order, the following Elementary Treatise is composed; which we have chosen rather to give in a Volume that is within the reach of every Student, than in one more pompous, which might be less generally useful. And we hope, from the pains it has cost us, its blemishes are not many, Ror such as a candid intelligent Reader may not forgive.

The Latin Appendix * is a proper Sequel, and a high Improvement, of what had been demonstrated in. Part III. concerning the Relation of Curve Lines and Equations ; a Subject which our Author had been early and intimately acquainted with; witness bis Geometria Organica, printed in 1719, when he was not full twenty-one years of age, and wbich, though so juvenile a work, gained him, at * A Translation of which is now given to this Edition, by the Rev.

Mr. Lawson.

once,

out

once, that diftinguished Rank among Mathematicians, which he thenceforth held with so great luftre. ret be frankly owns, be was led to many of the Propofitions in this Appendix, from a Theorem of Mr. Cores, communicated to bim, with

any demonstration, by the Reverend and Learned Dr. Smith, Master of Trinity College, Cambridge.. How be bas profited of that Hint, the Learned will judge: Thus much we can venture to say, that be bimself set some value upon this Performance ; baving, we are told, employed some of the latest bours could give to fucb Studies, in revising it for the Press; to bequeath it as bis last Legacy to the Sciences and to the Public..

Tbe gentlemen to cohóm Mr. MACLAURIN left the care of bis Papers, are MARTIN FOLKES, Esq. President of the Royal Society; ANDREW Mitchel, Esg. and the Reverend Mr. Hill, Chaplain to his Grace the Archbishop of Canterbury; with whom he had lived in a most intimate friendship. And by their direction this Treatise is published.

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