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NOTES AND OBSERVATIONS.
DEF. I. Book I. THE definition of a solid, contrary to the usual me. thod, is here made the first of the first Book; as those of a point, line and superficies are all derived from it, and cannot be understood without it. UCLID seems to have placed it in the eleventh book of the Elements, for the sake of uniformity; but arrangements of this kind, which are merely arbitrary, are but of little consequence, and should therefore always be made to give place to perfpicuity and the natural order of things.
DE F, 2, 3,-4. Book I. These definitions are now, by means of the former, rendered perfectly clear and intelligible, so that any farthere lucidation of them is altogether unnecessary. DR. SIMSON has endeavoured to Thew, by a formal proof, drawn from the confideration of a folid, that a point, according to Euclid's definition, is without parts, a line without breadth, and a furface without thickness; but this, and all other demonftrations of the fame kind, are unscientific and superfluous ; for these properties are so obviously effential to the things defined, that they cannot, even in idea, be feparated from them. If a point had parts, it would be a line; if a line had breadth it would be a superficies; and if a superficies had thickness, it would be a solid ; which are all manifest contradictions. It is, Þefides, a sure sign that a definition is badly expressed,
when it requires a number of prolix arguments to establish its truth and propriety.
DE F. 5. Book I. Euclid's definition of a right line is not expressed in so accurate and scientific a manner as could be wished; the lying evenly between its extreme points, is too vague and indefinite a term to be used in a science so much celebrated for its strictness and simplicity as Geometry. ARCHIMEDEs defines it to be the shortest diftance between any two points; but this is equally exceptionable, on account of the uncertain signification of the word distance, which, in common language, admits of various meanings. That which is here given, is, perhaps, not much preferable to either of these. The term right, or straight line, is, indeed, so common and fimple, that it seems to convey its own meaning, in a more clear and fatisfactory manner, than any explanation which can be given of it. DR. AUSTIN, in his Examination of the first fix books of the Elements, proposes a singular emendation of this definition, which includes the confideration of right lines, instead of a right line, as the case manifestly requires.
DEF. 6. Book I. Some call a plane superficies that which is the least of all those having the fame bounds; and others, that which is generated by the motion of a right line, not moving in the direction of itself; but these definitions are too complex and obscure to answer the purpose required. . Euclis defines it to be that which lies evenly between its lines, which is liable to the fame exceptions as that given of a right line ; nor is the one which has been substituted
in the place of this, by DR. SIMson, and other Editors, so simple and perspicuous as could be wished. Nothing is gained by the explanation of a term, if the words in which it is expressed are equally, or more, ambiguous, than the term itself: for this reason, that which is here given, has been preferred to either of those abovementioned; though, perhaps, it may not be equally commodious in certain cafes.
It is also to be remarked, that EUCLID never defines one thing by the intervention of another, as is the case in DR. SIMson's emendation; so that if this method had occured to him, he would certainly have rejected it.
DeF. 7. Book I. The general definition of an angle in Euclid, has been properly objected to, by several of the modern Editors, as being unneceffary, and conveying no distinct meaning; and in DR. SIMSON's emendation of the ninth, there feems to be still a superfluous condition. He delines a rectilineal angle, to be " the inclination of two straight lines to one another, which meet together, but are not in the same straight line .” Now their not being in the fame straight line, is a neceffary consequence, obviously included in their having an inclination to each other; and, therefore, to make this an essential part of the defini. tion, is certainly improper, and unscientific.
DEF, 8, 9. Book I. Euclid includes a right angle and a perpendicular in the same definition, which appears to be immethodical, and contrary to his usual custom. They are certainly distinct things, though dependent upon each other, and have as much claim to be separately defined, as a circle and its diameter,
DEF. 13. Book I. The definition of a circle from its generation, haş been thought by Dr. Barrow and others, to be preferable to Euclid's, or the one here given; as it is supposed to furnith its properties more readily, and to have the still farther advantage of shewing the actual existence of such 2 figure, independent of any hypothesis, but that of granting the poffibility of motion. But the requisition of this postulatum, appears to be a sufficient reason why Euclid rejected such a definition. The principles of pure Geometry, have no dependence upon motion, and it is, therefore, never used in the Elements, but in two or three places of the eleventh book, where it could not, without much obscurity and circumlocution, have been easily avoided. It is, besides, neither so simple, nor convenient to refer to, as Euclid's; which, in these respects, is as commodious as could be withed.
BOOK I. DR. BARROW, and other writers of considerable eminence, have censured Euclid for defining parallel lines, from the negative property of their never meeting each other; and to this they attribute all the perplexity and confusion, which has hitherto attended this delicate subject : affirming it as an utter impossibility, that any of the properties of these lines, can be derived from a definition which contains only a fimple negation. But these assertions appear to be groundless ; for the definition is founded on one of the most familiar, simple and obvious. properties of parallel lines, which either reason or science can discover: and, on this account, it is certainly preferable to any other that could have been formed from more abstruse and complicated affections of those lines, how ready and ufeful foever such a definition might have been found in its application.
The assertion, likewise, that none of the other properties of parallel lines can be derived from this definition, has been unadvisedly made ; for the 27th Prop. of the first Element, which is the same as the 22d of the present performance, is fairly and elegantly demonstrated by it; and by means something similar to those made use of by Dr. SIMSON, in his Notes upon the 29th Prop. it would not be difficult to shew that all the other properties of those lines may be derived from this definition, without the affistance of the 12th axiom, or any other of the same kind. Dr. Simson, indeed, in his attempt to demonstrate this axiom, has made several paralogisms which render his reasonings altogether inyalid, and nugatory. Passing by others, of less consequence, it will be sufficient to observe, that in his fifth Prop. he takes it for granted, that a line, which is per-pendicular to one of two parallel lines, may be produced till it meets the other: now this is a particular case of the very thing he is endeavouring to prove, which is so strange an oversight, that it is reinarkable how it could escape his observation.
This, however, is not the only instance of an unsuccessful attempt to prove the truth of the 12th axiom; for CLAVIUS and others have committed similar miltakes, and Dr. AUSTIN, who has endeavoured to demonstrate it by means of a new definition of parallel