"SELF-EXAMINATIONS IN ALGEBRA," &c. &c. CAMBRIDGE: W. P. GRANT, AND THE OTHER BOOKSELLERS; LONDON. 1829. PREFACE. THE present Work contains simply the results of experience. During a long series of Geometrical instructions to pupils of various capacities, many ideas have naturally suggested themselves, which the Author, considering valuable, has thought fit to embody in the following Treatise. His primary object has been, to render Euclid easier and more perspicuous; the next, to remedy the defects, and supply the omissions, by which it is vitiated; and the last, to continue the subject of Geometry up to the present advanced state of general Science. Different Pupils require different helps; one, even of inferior intellect, comprehending a Proposition of more real difficulty, in some instances, with much less trouble than another, of stronger talent, will make out an easier Proposition. It also happens, that the same Pupil at different times requires different assistance; and it may be assumed as a fact generally admitted, that explanations should be various, since what may be obscure even to darkness in one view, may be perfectly clear in another. These considerations have induced the Author to propose a number of Questions relative to the more difficult Propositions, such as he has been in the habit of giving to 427514 his Pupils, and which he apprehends are such as most Tutors find it necessary to give. Euclid, although decidedly the most perfect system of reasoning extant, is yet, in many respects, deficient or erroneous. These defects and errors have been explained, and, in most cases, removed. The text of Simson being taken as the standard, such oversights of that very learned, though partial Editor, as were palpable, have been freely noticed. ANALYTICAL GEOMETRY having been of late years very properly encouraged in the University, by way of APPENDIX TO THE SELF EXAMINATIONS OF THE TEXT, an introduction to that delightful and comprehensive subject is given. Referring all Points, Lines, Planes, Surfaces, and Solids, or whatever is the subject of Phlilosophical inquiry, to a certain fixed and definite position, it is commodious to imagine some fixed Point; then, passing through this Point, we conceive three Planes at Right Angles to one another, and perfectly unalterable in position. The positions relative to these Planes of any Magnitude being then given, the position of the Magnitude itself will be given. These Planes, called RECTANGULAR CO-ORDINATE PLANES, are of the most extensive use in Physics, the Positions, and therefore the Motions, or Changes of Positions, of all Bodies, being referred to such Planes in all the best modern Works on Astronomy, Mechanics, and the other branches of the Applied Mathematics. These general views of Science, first introduced by Maclaurin, and subsequently adopted by all the Philosophers of the Continent, and the more enlightened of this Country, have given rise to the cultivation of ANALYTICAL GEOMETRY according to the method of RECTANGULAR CO-ORDINATES. It is in subserviency to this system of reference of all Points to RECTANGULAR COORDINATE PLANES, that the first APPENDIX has been added.* This APPENDIX is followed by the famous TREATISE OF PAPPUS ON TANGENCIES, as restored by VIETA. This is extracted from a Work by LAWSON, which is now very scarce, and not likely to be reprinted. The demonstrations are left incomplete, in order to exercise the ingenuity of the Student. Another APPENDIX, containing DEDUCTIONS FROM EUCLID, has been given, It was the Author's intention at first to have made a Selection of such only as are of use in Physics and Natural Philosophy, and the other branches of Geometry, such as Conics, Spherics, &c. But a slight inspection of the best Works on these branches was sufficient to shew, that much as "Deductions" are cultivated in the University, and despite of the thousands which have teemed from the Press, very few * If the reader choose to apply to Mons. Picnot, of 15, Suffolk Street, Pall-Mall East, London, he may be furnished, at a moderate expense, with a set of Co-ordinate Planes, intersecting each other at right angles, and meeting at a common origin. This ingenious artist devised for the Author and a Pupil of his, a mode of actually exhibiting the Co-ordinates of a Point in Space; of Planes, Spheres, Cones, and other Solids in Space, by means of a joint moveable in all directions, something like the Shoulder-joint, which will place the Point, Plane, or Solid, in discussion, in any position that may be required. Students, when furnished with this apparatus, will find the ordinary difficulties attending the representations in solid Geometry, all vanish. |