A TREATISE OF ALGEBRA: WHEREIN THE PRINCIPLES ARE DEMONSTRATED, AND APPLIED IN MANY USEFUL AND INTERESTING INQUIRIES, AND IN THE RESOLUTION OF A GREAT VARIETY OF PROBLEMS OF TO WHICH IS ADDED, THE GEOMETRICAL CONSTRUCTION OF A GREAT NUM- With the Method of Resolving the same Numerically. BY THOMAS SIMPSON, F. R. S. THE SECOND AMERICAN, FROM THE EIGHTH LONDON EDITION. PRINTED FOR M. CAREY & SONS, DAVID HOGAN, BENNETT & WALTON, AND ABRAHAM SMALL. EASTERN DISTRICT OF PENNSYLVANIA, TO WIT: BE IT REMEMBERED, That on the seventh day of Decem. (L. S.) ber, in the forty-sixth year of the Independence of the United States of America, A. D. 1821, Mathew Carey & Sons, of the said District, have deposited in this office the Title of a Book, the right whereof they claim as Proprietors, in the words following, to wit: "A Treatise of Algebra: wherein the principles are demonstrated, "and applied in many useful and interesting inquiries, and in the reso"lution of a great variety of Problems of different kinds. To which is "added, the Geometrical Construction of a great number of linear and "plane Problems, with the method of resolving the same numerically. "By Thomas Simpson, F. R. S. The second American, from the eighth "London edition. Revised, corrected and improved, by David M'Clure, "Teacher of the Mathematics." In conformity to the Act of the Congress of the United States, intituled, "An Act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned." And also to the Act, entitled, "An Act supplementary to an Act, entitled, 'An Act for the encourage. ment of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned,' and extending the benefits thereof to the arts of designing, engraving, and etching historical and other prints." D. CALDWELL, Clerk of the Eastern District of Pennsylvania. THE motives that first gave birth to the ensuing work, were not so much any extravagant hopes the author could form to himself of greatly extending the subject, by the addition of a large variety of new improvements, (though the reader will find many things here that are nowhere else to be met with,) as an earnest desire to see a subject of such importance, established on a clear and rational foundation, and treated as a science, capable of demonstration, and not a mysterious art, as some authors, themselves, have thought proper to term it. How well the design has been executed, must be left for others to determine. It is possible that the pains here taken, to reduce the fundamental principles, as well as the more difficult parts of the subject, to a demonstration, may be looked upon, by some, as rather tending to throw new difficulties in the way of a learner, than to the facilitating of his progress. In or der to gratify, as far as might be, the inclination of this class of readers, the demonstrations are now given by themselves, in the manner of notes, (so as to be taken or omitted at pleasure:) though the author cannot by any means be induced to think that time lost to a learner, which is taken up in comprehending the grounds whereon he is to raise his superstructure: his |