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treats of, and the space contained within it, are problems that certainly belong to the elements of the science, especially as they are not more difficult than other propofitions where the method of exhaustions is employed. When I speak of the rectification of the circle, or of measuring the length of the circumference, I must not be supposed to mean, that a straight line is to be made equal to the circumference exactly, a problem which, as is well known, Geometry has never been able to resolve : All that is proposed is, to determine two traight lines that differ very little from one another, not more, for instance, than the four hundred and ninety-seventh part of the diameter of the circle, and of which the one is demonstrated to be greater than the circumference of that circle, and the other to be less. In the same manner, the quadrature of the circle is performed only by approximation, or by finding two rectangles, nearly equal to one another, the one of them greater, and the other less than the space contained within the circle.
The Data of Euclid has been annexed to fe. veral editions of the 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 itself an analytical form ; and, for these reasons, I would willingly reserve it, or rather a compend of it, for a work on that analysis, which I have long meditated.
Plane and Spherical Trigonometry, on the other hand, make a part of this volume, because, ih
every course of mathematical studies, that is directed toward useful purpofes, these two branches neceffarily come after the Elements. In explaining the elements of such sciences, 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, though pretty full, is so divided, that the part of it that is barely sufficient for the resolution of Triangles, may be easily taught by itself. In a fcholium the method of constructing the trigonometrical Tables is explained, and a demonstration is added of the properties of the fines and co-fines of the fums and differences of arches, which are the foundation of those new applications of Trigonometry that have been introduced with so 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 the new tables of Logarithms.
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 propofitions, which every body is ready to admit, without any demonstration, and thus take up the time, and fatigue the attention of the student, to no purpose. To this objection also, if there be any force in it, the present treatise is certainly as much exposed as any other, for, of all the alterations that may be made in the Elements, the last I should think of, is to consider any thing as self-evident that admits of demonftration. 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 propofition, but to shew its necessary connection with other propofitions, 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 exifts. It is uponthis that the beauty and peculiar excellence of the mathematical sciences depend; it is this that prevents any one truth from being single and insulated, and connects the different parts fo firmly, that they must all stand, or all fall together. The demonstration, therefore, even of an obvious propofition, 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 would cease to be true if this proposition were supposed to be falfe? 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 oppofite to the greater angle, nor that the equal fides are opposite to equal angles, nor, lastly, that things equal to the fame 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 reasoning, cannot be employed to greater advantage than in analysing those judgments, which, though they appear fimple, are in reality complex, and capable of being diftinguished into parts. No progress in ascending higher can be expected till a regular habit of demonstration is thus acquired ; and I should greatly suspect, that he who has declined the trouble of tracing the connection between the proposition already quoted, and those that are below it, would never be very expert in tracing its connection with those that are above it; and that, as he had not been careful in laying the foundation, he would never be successful in raising the superstructure.
COLLEGE OF EDINBURGH,
Oct. 21. 1795