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5th Book, by which the doctrine of compound ratios is rendered plain and easy. Besides, among the Definitions of the 11th Book, there is this, which is the 10th, viz. "Equal and "similar solid figures are those which are contained by similar "planes of the same number and magnitude." Now this Proposition is a Theorem, not a Definition; because the equality of figures of any kind must be demonstrated, and not assumed; and therefore, though this were a true Proposition, it ought to have been demonstrated. But, indeed, this Proposition, which makes the 10th Definition of the 11th Book, is not true universally, except in the case in which each of the solid angles of the figures is contained by no more than three plane angles; for in other cases, two solid figures may be contained by similar planes of the same number and magnitude, and yet be unequal to one another, as shall be made evident in the notes subjoined to these Elements. In like manner, in the Demonstration of the 26th Prop. of the 11th Book, it is taken for granted, that those solid angles are equal to one another which are contained by plane angles of the same number and magnitude, placed in the same order; but neither is this universally true, except in the case in which the solid angles are contained by no more than three plane angles; nor of this case is there any demonstration in the Elements we now have, though it be quite necessary there should be one. Now, upon the 10th Definition of this Book depend the 25th and 28th Propositions of it; and upon the 25th and 26th depend other eight, viz. the 27th, 31st, 32d, 33d, 34th, 36th, 37th, and 40th of the same Book; and the 12th of the 12th Book depends upon the 8th of the same; and this eighth, and the Corollary of Proposition 17. and Proposition 18. of the 12th Book, depend upon the 9th Definition of the 11th Book, which is not a right definition; because there may be solids contained by the same number of similar plane figures, which are not similar to one another, in the true sense of similarity received by Geometers; and all these Propositions have, for these reasons, been insufficiently demonstrated since Theon's time hitherto. Besides, there are several other things, which have nothing of Euclid's accuracy, and which plainly show, that his Elements have been much corrupted by unskilful Geometers, and, though these were not so gross as the others now mentioned, they ought by no means to remain uncorrected.
Upon these accounts it appeared necessary, and I hope will prove acceptable, to all lovers of accurate reasoning, and of mathematical learning, to remove such blemishes, and restore the principal Books of the Elements to their original accuracy
as far as I was able; especially since these Elements are the foundation of a science by which the investigation and discovery of useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewise is of the greatest use in the arts both of peace and war, to many of which Geometry is absolutely necessary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of Geometers, and by restoring to him those things which Theon or others have suppressed, and which have these many ages been buried in oblivion.
In this edition, Ptolemy's Proposition concerning a property of quadrilateral figures in a circle is added at the end of the sixth Book. Also the note on the 29th Proposition, Book 1. is altered, and made more explicit, and a more general demonstration is given, instead of that which was in the Note on the 10th Definition of Book 11.; besides the Translation is much amended by the friendly assistance of a learned gentleman.
To which are also added, the Elements of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid.
TO THIS EDITION.
ALTHOUGH various attempts have been made to simplify and improve the Elements of Euclid, yet Dr Simson's edition still deservedly remains a standard.
This is evident from the very favourable reception which the former editions of this work have met with from the Public, and from its general introduction into Mathematical Seminaries. In order to secure a continuance of the public approbation, the Proprietors have been at very great pains and additional expense on this new impression, which has been every where most carefully revised and corrected.
The Editor hopes that, by the additions which he has made to the Plane and Spherical Trigonometry, the main objections, advanced against that part of the work in former editions, are now obviated; and he confidently trusts that the present impression will be found more accurate and complete than any of those which have preceded it.
EDINBURGH, 6th July 1835.
ELEMENTS OF EUCLID.
I. A point is that which hath no parts, or which hath no mag- Book I.
II. A line is length without breadth.
III. The extremities of a line are points.
IV. A straight line is that which lies evenly between its extreme points.
V. A superficies is that which hath only length and breadth.
VI. The extremities of a superficies are lines.
VII. A plain superficies is that in which any two points being See N. taken, the straight line between them lies wholly in that superficies.
VIII. "A plane angle is the inclination of two lines to one See N. "another in a plane, which meet together, but are not in "the same direction."
IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
N. B. When several angles are at one point B, any one ⚫ of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line: Thus the angle which is contained by the straight lines AB, CB is named the angle ABC, or CBA; that which is contained by AB, BD is named the angle ABD, or DBA ; and that which is contained by BD, CB is called the angle DBC, or CBD: but if there be only one angle at 'a point, it may be expressed by a letter placed at that point; as the angle at E.'
X. When a straight line standing on
XI. An obtuse angle is that which is greater than a right angle.
XII. An acute angle is that which is less than a right angle.