RIGHT HONOURABLE JAMES EARL of MORTON, Lord ABERDOUR, Knight of the most ancient Order of the THISTLE, One of the Sixteen Peers for SCOTLAND, Vice-Admiral of ORKNEY and ZETLAND, Prefident of the Philofophical Society at EDINBURGH, and Fellow of the Royal Society of LONDON. MY LORD, YOUR OUR Character will be a fufficient apology for my defiring the honour to infcribe the following Sheets to your Lordship, and your Goodness will pardon the liberty I take, as it affords me an opportunity of teftifying the high refpect and esteem with which I am, MY LORD, Your Lordship's most devoted, most obedient, and most humble servant, Thomas Simpson. THE AUTHOR's PREFACE TO THE SECOND EDITION. T HE motives that first gave birth to the enfuing Work, were not fo much any extravagant hopes the Author could form to himself of greatly extending the fubject by the addition of a large variety of new improvements (though the Reader will find many things here that are no where elfe to be met with) as an earneft defire to fee á fubject of fuch general importance established on a clear and rational foundation, and treated as a fcience, capable of demonftration, and not a mysterious art, as fome authors, themfelves, have thought proper to term it. How well the defign has been executed, muft be left for others to determine. It is poffible that the pains here taken, to reduce the fundamental principles, as well as the more difficult parts of the fubject to a demonstration, may be looked upon, by fome, as rather tending to throw new difficulties in the way of a Learner, than to the facilitating of his progrefs. In order to gratify, as far as might be, the inclination of this clafs of Readers, the demonstrations are now given by themfelves, in the manner of Notes Gift Prof. ww. Beman 4-12-33 PREFACE. Notes (fo as to be taken or omitted at pleafure); though the Author, cannot by any means be induced to think, that Time loft to a Learner which is taken up in comprehending the grounds whereon he is to raise his fuperftructure: his progrefs may indeed, at first be a little retarded; but the real knowledge he thence acquires will abundantly compenfate his trouble, and enable him to proceed, afterwards, with certainty and fuccefs, in matters of greater difficulty, where authors, and their rules, can yield him no affiftance, and he has nothing to depend upon but his own obfervation and judgment. This, fecond, Edition has many advantages over the former, as well with refpect to a number of new fubjects and improvements, inter fperfed throughout the whole, as in the order and difpofition of the elementary parts: in which particular regard has been had to the capacities of young Beginners. The Work, as it now ftands, will, the Author flatters himself, be found equally plain and comprehenfive, fo as to anfwer, alike, the purpose of the lower, and of the more experienced clafs of Readers. THE I. Of the origin and compofition of Equations 2. How to know whether fome, or all the roots of an Equation be rational, and, if fo, what they are 134 3. Another way of discovering the fame thing,by means 7. The folution of biquadratics by a new method with- out the trouble of exterminating the fecond term 150 8. Cafes of biquadratic Equations that may be re- 9. The refolution of literal Equations, wherein the |