T existence of such solids, or such lines as are evi dently possible, though the manner of actually de scribing them may not have been explained. In this way, I have been enabled to offer that very refined artifice in geometrical reasoning, to which we give the name of the Method of Exhaustions, under a much simpler form than it appears in the 12th of EUCLID; and the spirit of the method may, I think, be best learned, when it is thus disengaged from every thi not essential. That it may be the better understood, and because the demonstrations which require exhaustions, are, no doubt, the most difficult in the Elements, they are all conducted as nearly as possible in the same way, in the cases of the different solids, from the pyramid to the sphere. The comparison of this last solid with the cylinder, concludes the last Book of the Supplement, and is a proposition that may not improperly be considered as terminating the elementary part of wrtentos deid eft eti a dor Geometry. rit The Book of the Data has been annexed to se veral editions of EUCLID'S Elements, and particularly to Dr SIMSON's, but in this it is omitted altogether. It is omitted, however, not from any opinion of its being in itself useless, but because it does not belong to this place, and is not often read by beginners. It contains the rudiments of what is properly called the Geometrical Analysis, and has in itself an analytical form; and for these rea sons I would willingly reserve it, or rather a compend of it, for a separate work, intended as an introduction to the study of that analysis.onfor In explaining the elements of Plane and Spherical Trigonometry, there is not much new that can be attempted, or that will be expected by the intelligent reader. Except, perhaps, some new demonstrations, and some changes in the arrangement, these two treatises have, accordingly, no novelty to boast of The Plane Trigonometry is so divided, that the part of it that is barely sufficient for the resolution of Triangles, may be easily taught by itself. The method of constructing the Trigonometrical Tables is explained, and a demonstration is added of those properties of the sines and cosines of arches, which are the foundation of those applications of Trigonometry lately introduced, with s much advantage, into the higher Geometry In the Spherical Trigonometry, the rules for preventing the ambiguity of the solutions, whereever it can be prevented, have been particularly attended to; and I have availed myself as much as possible of that excellent abstract of the rules of this science, which Dr MASKELYNE has prefixed to TAYLOR'S Tables of Logarithms, An explanation of NAPIER's very ingenious and useful rule of the Circular Parts is here added as an appendix to Spherical Trigonometry. It has been objected to many of the writers on Elementary Geometry, and particularly to EUCLID, that they have been at great pains to prove the truth of many simple propositions, which every body is ready to admit, without any demonstration, and that they thus take up the time, and fatigue the attention of the student, to no purpose. To this objection, if there be any force in it, the present treatise is certainly as much exposed as any other; for no attempt is here made to abridge the Elements, by considering as self-evident any thing that admits of being proved. Indeed those who make the objection just stated, do not seem to have reflected sufficiently on the end of Mathematical Demonstration, which is not only to prove the truth of a certain proposition, but to shew its. necessary connection with other propositions, and its dependence on them. The truths of Geometry are all necessarily connected with one another, and the system of such truths can never be rightly explained, unless that connection be accurately traced, wherever it exists. It is upon this that the beauty and peculiar excellence of the mathematical sciences depend: it is this, which, by preventing any one truth from being single and insulated, connects the different parts so firmly, that they must all stand or all fall together. The demonstration, therefore, even of an obvious proposi tion, answers the purpose of connecting that proposition with others, and ascertaining its place in the general system of mathematical truth. If, for example, it be alleged, that it is needless to demonstrate that any two sides of a triangle are greater than the third; it may be replied, that this is no doubt a truth, which, without proof, most men will be inclined to admit; but, are we for that reason, to account it of no consequence to know what the propositions are, which must cease to be true if this proposition were supposed to be false? Is it not useful to know, that unless it be true, that any two sides of a triangle are greater than the third, neither could it be true, that the greater side of every triangle is opposite to the greater angle, nor that the equal sides are opposite to equal angles; nor, lastly, that things equal to the same thing are equal to one another? By a scientific mind, this information will not be thought lightly of; and it is exactly that which we receive from EUCLID's demonstration. To all this, it may be added, that the mind, especially when beginning to study the art of reason ing, cannot be employed to greater advantage than in analysing those judgments, which, though they appear simple, are in reality complex, and capable of being distinguished into parts. No progress in ascending to higher investigations can be expected, till a regular habit of demonstration in elementary matters has been acquired; it is therefore to be feared, that he who has declined the trouble of tracing the connection between propositions such as that already quoted, and those that are more simple, will not be very expert in tracing their connec tion with those which are more complex; and that, as he has not been careful in laying the foundation, he will never be successful in raising the superstructure. COLLEGE OF EDINBURGH, } ADVERTISEMENT TO THE SIXTH EDITION. The Fifth Edition of the late Professor PLAYFAIR'S Elements of Geometry, and the last that had the benefit of his own superintendance, having been exhausted, this New Edition has been printed under my care. COLLEGE OF EDINBURGH, WILLIAM WALLACE, |