prefent mode of expreffion is very nearly the fame with that of the former, and the intention and ufe are exactly the fame. In the Fifth Book, however, the change of expreffion made in the Definitions, caufes a fimilar change in their application, on which account, in the Demonftrations, there is fometimes a different ftep neceffary in order to connect them with the Definitions, and fometimes a difference in the conftruction, but it is generally made more fimple than before. Befides, in this Book, the form of the constructions is altered, the multiples being now exhibited, by increafing the magnitudes, instead of being made different magnitudes, as they were before; and thofe of them that are equimultiples, are marked with the fame letters: By which means, their dependence upon their magnitudes will be more evident, and the Student will find no difficulty, either in difcovering the multiples of magnitudes, or in knowing which of them are equimultiples;-things which created confiderable trouble before. In other refpects, this Book is the fame as before, except that the 1ft, 2d, and 6th Propofitions are more general, and that the Demonftrations near the beginning are very fully expreffed.

It will be shown in the Notes, that the Definition of proportionals now given, is almoft the fame with the ancient definition; and it is obvious, that it agrees with the modern definition, and is a much better expreffion of it, than that which is commonly given and it is as easily applied as either of them to the purpose of demonftrating the properties of proportionals; so that there does not appear to be any valid objection against it.

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

ufeful Propofitions to it: but this defign is dropt at prefent, because that Book is very prolix, and feldom read by beginners; and the additional Propofitions could not be easily deduced from it; and to demonftrate them, independent of it, would have fwelled the Book too much. It is therefore thought to be more convenient, efpecially for beginners, for whose ufe this Book is chiefly intended, to demonftrate the relations of the parallelopiped and prifm to the folids, which are the fubject of this Book, and from them to deduce the principal Propofitions of the 12th Book, which cafily flow from them; thus forming a plain and fhort abridgement of it... In the conftructions of this Book, the figures inscribed in the circles are compofed of rectangles made in the manner of the moderns, but the Demonftrations 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 exhauftions, does not lie in the methods themselves, but in the inaccurate mode of expreffion ufed 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 5th, and its corollaries; and the 10th, 11th, and 12th, in the 6th and 7th, and their corollaries; and the 18th is the 10th of this; all the other Propofitions of this Book of Euclid are only fubfidiary ones.

Besides thefe alterations, there are several parti-cular errors corrected in this Work, and many of the Demonftrations which were formerly given in different cafes, are now made more general, and many others are fhortened. Likewife, a number of ufeful Definitions and Propofitions are added to the Elements, and fome ufelefs ones thrown out. But

for

for a more particular account of thefe 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 demonftrated without any affumption, and the other Axioms, which the moderns have attempted to fubftitute for it, are particularly confidered: and in the Notes on the Fifth Book, the Definition of proportionals is deduced from the manner of obtaining our firft ideas of proportion.

In the Elements of Plane and Spherical Trigonometry, annexed to fome of the Editions of DrSimfon's Euclid, feveral things were affumed without proof, which gave confiderable 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. Likewife, the nature and ufe of the Trigonometrical Tables are explained after the 2d Propofition, and their conftruction 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 refolving the cafes, 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 demonftrated, as any of the particular ones; and eafy rules are given for preventing the ambiguity of the Solutions. There are also many new Demonftrations given in Trigonometry, much more fimple than the former ones.

It is acknowledged, that in this performance the

brevity

brevity affected by fome modern writers is indu ftriously avoided, because it is well known, from experience, that inftead of furthering, it greatly retards the progrefs of Students; and for the fame reafon, Algebraic fymbols are avoided. The Editor is of the fame opinion with Dr Keil, that "the Elements of all Sciences ought to be handled in the moft fimple manner, and not to be involved in Symbols, Notes, or obfcure Principles."

:

But though words be not used fo fparingly here as by fome others, there are very few that could be wanted, without producing obfcurity and as to tediousness, so often complained of, it is rather in appearance than reality; for the arrangement is so happily contrived, as almoft always to admit the fimpleft conftructions and demonftrations 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 lofs to Geometry. Nor is the method of handling folids in the 11th Book fo tedious as is alledged: for if we omit the 22d, 23d, 26th, and 27th, together with all thofe after the 34th, as is ufually done by the moderns, the reft are ftill accurately demonftrated without them; and thus, the number of the Propofitions concerning fo. lids is reduced to ten, with two Corollaries, a number as fmall as has ever been used, or can reasonably be expected in treating fuch a copious fubject.

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

By Practical Geometry is here meant, the method of expreffing the magnitude of lines, fuperficies, &c. by means of the measures in coinmon ufe, fuch as

inches,

inches, feet, &c. for which purpose, it is affumed, that the magnitudes concerned are all commenfurable, or rather, that they exceed commenfurable magnitudes, by differences too inconfiderable to be taken notice of; so that, when a measure is applied, for example, to a line, that line is fuppofed to contain the measure, or fome part of it, a certain number of times, without any remainder deferving notice. In this Treatife, the Demonftrations are as accurate as in the Elements, but they are not quite fo full, because the Student is now better acquainted with the nature of Demonftration, and its peculiar mode of expreffion, than when he was reading the Elements. It is very fhort, being only an introduction to Menfuration, Surveying, Guaging, &c. and is not intended to fuperfede the perufal of more complete Treatifes on these fubjects.

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