fionally improving to the Perfection be intended it Should have, before be committed it to the Prefs. And the best Copies of his Manufcript having been tranfmitted to the Publisher, it was eafy, by comparing them, to establish a correct and genuine Text. There were, befides, feveral detached Papers, fome of which were quite finished, and wanted only to be inferted in their proper places. In a few others, the Demonftrations were fo concisely expreffed, and couched in Algebraical characters, that it was neceflary to write them out at more length, to make them of a piece with the rest... And this is the only liberty the Publifher has allowed bimfelf to take; excepting a few inconfide-. rable additions, that feemed neceffary to render the Book more complete within itself, and to fave the trouble of confulting others who have written on the fame Subject. The Rules concerning the Impoffible Roots of Equations, our Author had very fully confidered,. as appears from his Manufcript Papers: but as he had no where reduced any thing on that Subject to a better form, than what was long ago publifhed in the Philofophical Tranfactions, No. 394 and 408, we thought it best to take the fubftance of Chap. 11. Part II. from thence; efpecially as the latter of thefe Papers furnishes a demonstration of the original Rule, which pre-supposes only what the Reader has been taught in a preceding Chapter. The Paper that is fubjoined, 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 fome things added by his Lordship, which were wanting to finish the Demonftration. Of thefe Materials, carefully collected and put in order, the following Elementary Treatife is compofed; which we have chofen rather to give in a Volume that is within the reach of every Student, than in one more pompous, which might be lefs generally useful. And we hope, from the pains it has coft us, its blemishes are not many, nor fuch as a candid intelligent Reader may not forgive. * The Latin Appendix is a proper Sequel, and abigh Improvement, of what had been demonftrated in Part III. concerning the Relation of Curve Lines and Equations; a Subject which 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 so juvenile a work, gained him, at A Tranflation of which is now given to this Edition, by the Rev. Mr. Lawfon. once, once, that diftinguished Rank among Mathemati cians, which be thenceforth held with fo 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 Trinity College, Cambridge. How he has profited of that Hint, the Learned will judge: Thus much we can venture to fay, that he himself fet fome value upon this Performance; having, we are told, employed fome of the latest hours be could give to fuch Studies, in revifing it for the Press; to bequeath it as his laft Legacy to the Sciences and to the Public.. 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 most intimate friendship. And by their direction this Treatife is published.. CON. XI. Of the Solution of queftions that pro- |