treats of, and the space contained within it, are problems that certainly belong to the elements of the fcience, efpecially 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 fuppofed 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 refolve: All that is propofed is, to determine two ftraight lines that differ very little from one another, not more, for inftance, than the four hundred and ninety-feventh part of the diameter of the circle, and of which the one is demonftrated to be greater than the circumference of that circle, and the other to be less. In the fame 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 lefs than the space contained within the circle.

The Data of EUCLID has been annexed to feveral 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

called the Geometrical Analysis, and has itself an analytical form; and, for these reasons, I would willingly referve 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, in every course of mathematical ftudies, that is directed toward useful purposes, these two branches neceffarily come after the Elements. In explaining the elements of fuch fciences, there is not much new that can be attempted, or that will be expected by the intelligent reader. Except, perhaps, fome new demonftrations, and fome changes in the arrangement, these two treatises have, accordingly, no novelty to boast of. The Plane Trigonometry, though pretty full, is fo divided, that the part of it that is barely fufficient for the refolution of Triangles, may be eafily taught by itself. In a fcholium the method of conftructing the trigonometrical Tables is explained, and a demonftration is added of the properties of the fines and co-fines of the fums and differences of arches, which are the foundation of thofe new applications of Trigonometry that have been introduced with fo much advantage into the higher Geometry.

In the Spherical Trigonometry, the rules for preventing the ambiguity of the folutions, whereever it can be prevented, have been particularly attended to; and I have availed myself as much as poffible of that excellent abftract 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 fimple 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 treatife is certainly as much expofed as any other, for, of all the alterations that may be made in the Elements, the laft I fhould think of, is to confider any thing as felf-evident that admits of demonstration. Indeed, thofe who make the objection juft ftated, do not feem to have reflected fufficiently on the end of Mathematical Demonstration, which is not only to prove the truth of a certain propofition, but to fhew its neceffary connection with other propofitions, and its dependence on them. The truths of Geometry are all neceffarily connected with one another, and the fyftem of fuch truths can never

be

be rightly explained, unless that connection be accurately traced, wherever it exists. It is uponthis that the beauty and peculiar excellence of the mathematical fciences depend; it is this that prevents any one truth from being fingle and infulated, 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 propofition with others, and afcertaining its place in the general fyftem of mathematical truth. If, for example, it be alleged, that it is needless to demonftrate that any two fides 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 confequence to know what the propofitions are, which would cease to be true if this propofition were fuppofed to be falfe? Is it not useful to know, that unless it be true, that any two fides of a triangle are greater than the third, neither could it be true, that the greater fide of every triangle is oppofite to the greater angle, nor that the equal fides are oppofite to equal angles, nor, laftly, that things equal to the fame thing are equal to one another? By a fcientific mind this information will not be thought lightly of; and it is exactly that which we receive from EUCLID'S demonftration.

Το

To all this it may be added, that the mind, especially when beginning to study the art of reafoning, cannot be employed to greater advantage than in analyfing thofe judgments, which, though they appear fimple, are in reality complex, and capable of being diftinguifhed into parts. No progrefs in ascending higher can be expected till a regular habit of demonftration is thus acquired; and I fhould greatly fufpect, that he who has declined the trouble of tracing the connection between the propofition 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 fuccefsful in raifing the fuperftructure.

COLLEGE OF EDINBURGH,

Oct. 21. 1795.