présent mode of expression is very nearly the same with that of the former, and the intention and use are exactly the same. In the Fifth Book, however, the change of expression made in the Definitions, causes a similar change in their application, on which account, in the Demonstrations, there is sometimes a different step necessary in order to connect them with the Definitions, and fometimes a difference in the construction, but it is generally made more simple than before. Besides, in this Book, the form of the constructions is altered, the in ultiples being now exhibited, by increasing the magnitudes, instead of being made different magnitudes, as they were before; and those of them that are equimultiples, are marked with the same letters: By which means, their dependence upon their magnitudes will be more evident, and the Student will find no difficulty, either in discovering the multiples of inagnitudes, or in knowing which of them are equimultiples ;--things which created considerable trouble before. In other respects, this Book is the same as before, except that the ist, 2d, and 6th Propositions are more general, and that the Demonstrations near the beginning are very fully expressed.

It will be shown in the Notes, that the Definition of proportionals now given, is almost the same with the ancient definition; and it is obvious, that it agrees with the modern definition, and is a much better expression of it, than that which is commonly given : and it is as easily applied as either of them to the purpose of demonstrating the properties of proportionals ; so that there does not appear to be

i any valid objection against it.

It was at first intended to have given the 12th Book of Euclid entire, and to have annexed fome A2

useful

useful Propositions to it: but this design is dropf ať present, because that Book is very prolis, and feldom read by beginners; and the additional Propofitions could not be easily deduced from it; and to demonstrate them, independent of it, would have swelled the Book too much, :, It is therefore thought to be more convenient, especially for beginners, for whose use this Book is chiefly intended, to demonstrate the relations of the parallelopiped and prifin to the folids, which are the fubje& of this Book, and from them to deduce the principal Propositions of the 12th Book, which casily flow from them ; thus forming a plain and thort abridgement of it. In the contructions of this Book, the figures inscribed in the circles are composed of rectangles made in the manner of the moderns, but the Demonstrations are conducted in the manner of the ancients : by which means it is manifeft, that the principal difference between the ancient and modern methods of exhaustions, does not lie in the methods themselves, but in the inaccurate mode of expression used by many of the moderns.

The 2d of the 12th Book of Euclid is the 3d of this; and the 5th, 6th, and 7th, are contained in the śth, and its corollaries; and the loth, lith, and

I 2th, in the 6th and 7th, and their corollaries ; and the 18th is the roth of this; all the other Propositions of this Book of Euclid are only subsidiary ones.

Besides these alterations, there are several parti. cular errors corrected in this work, and many of the Demonstrations which were formerly given in different cases, are now made more general, and many

others are shortened. Likewise, a number of useful Definitions and Propositions are added to the Elements, and some useless ones thrown out. But

for

for a more particular account of these things, the reader is referred to the Notes, in which he will also find the nature of Geometrical accuracy treated pretty fully; and in the Notes on the 29th Propofition of the First Book, the 12th Axiom is demonstrated without any affumption, and the other Axioms, which the moderns have attempted to substitute for it, are particularly considered and in the Notes on the Fifth Book, the Definition of proportionals is deduced from the manner of obtaining our first ideas of proportion.

In the Elements of Plane and Spherical Trigonometry,annexed to fome of the Editions of DrSimfon's Euclid, several things were affumed without proof, which

gave considerable trouble to beginners. These Elements are now made more accurate and complete ; and a new Lemma is prefixed, which is the foundation of the application of Arithmetic to Geometry. Likewise, the nature and ufe of the Trigonometrical Tables are explained after the 2d Propofition, and their construction is given at the end of Plane Trigonometry, to which also there is added a method of finding the ratio of the circumference of a circle to its diameter ; for without the knowledge of these things, Trigonometry cannot be fully understood. And in Spherical Trigonometry, the Proportions for resolving the cases, which were formerly fo numerous as to be a burden to the memory, , are now reduced to a few general ones, that are as easily understood and demonstrated, as any of the particular ones; and easy rules are given for

preventing the ambiguity of the Solutions. There are also many new Demonstrations given in Trigonometry, much more simple than the former ones. It is acknowledged, that in this performance the

brevity

brevity affected by some modern writers is indus striously avoided, because it is well known, from experience, that instead of furthering, it greatly retards the progress of Students; and for the same reason, Algebraic symbols are avoided. The Editor is of the same opinion with Dr Keil, that " the Elements of all Sciences ought to be handled in the most simple manner, and not to be involved in Symbols, Notes, or obscure Principles.”

But though words be not used so sparingly here as by fome others, there are very few that could be wanted, without producing obscurity: and as to tediousness, so often complained of, it is rather in appearance than reality; for the arrangement is fo happily contrived, as almost always to admit the simplest constructions and demonstrations that can be given: and in all the first Six Books, there are not above half a dozen of Propofitions that could be omitted, without a loss to Geometry. Nor is the method of handling solids in the rith Book so tedious as is alledged: for if we omit the 22d, 23d, 26th, and 27th, together with all those after the 34th, as is usually done by the moderns, the rest are still accurately demonstrated without them; and thus, the number of the Propositions concerning fo. lids is reduced to ten, with two Corollaries, a number as small as has ever been used, or can reasonably be expected in treating such a copious subject.

At the end, there is added a Treatise of Practical Geometry, a subject to which the attention of Students is almost always directed, immediately after they have read the Elements.

By Practical Geometry is here meant, the method of expressing the magnitude of lines, superficies, &c. by means of the measures in coinmon use, such as

inches,

inches, feet, &c. for which purpose, it is assumed, that the magnitudes concerned are all commensurable, or rather, that they exceed commensurable magnitudes, by differences too inconsiderable to be taken notice of; so that, when a measure is applied, for example, to a line, that line is suppoled to contain the measure, or some part of it, a certain number of times, without any remainder deferying notice. In this Treatise, the Demonstrations are as accurate, as in the Elements, but they are not quite so full, because the Student is now better acquainted with the nature of Demonstration, and its peculiar mode of expression, than when he was reading the Elements. It is very short, being only an introduction to Mensuration, Surveying, Guaging, &c. and is not intended to supersede the perusal of more complete Treatises on these subjects.

ER