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5th Book, by which the doctrine of compound ratios is rendered plain and eafy. Besides among the Definitions of the 11th Book, there is this, which is the 10th, viz. "Equal and "fimilar folid figures are thofe which are contained by fimilar "planes of the fame number and magnitude." Now this Propofition is a Theorem, not a Definition; because the equality of figures of any kind muft be demonftrated, and not affumed; and therefore, though this were a true Propofition, it ought to have been demonftrated. But, indeed, this Propofition, which makes the 10th Definition of the 11th Book, is not true univerfally, except in the cafe in which each of the folid angles of the figures is contained by no more than three plane angles; for in other cafes, two folid figures may be contained by fimilar planes of the fame number and magnitude, and yet be unequal to one another, as fhall be made evident in the Notes fubjoined to thefe Elements. In like manner, in the Demonftration 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 plain angles of the fame num ber and magnitude, placed in the fame order; but neither is this univerfally true, except in the cafe in which the folid angles are contained by no more than three plain angles; nor of this cafe is there any Demonftration in the Elements we now have, though it be quite neceffary there fhould be one. Now, upon the roth Definition of this Book depend the 25th and 28th Propofitions of it; and, upon the 25th and 26th depend other eight, viz. the 27th, 31ft, 32d, 33d, 34th, 36th, 37th, and 40th of the fame Book; and the 12th of the 12th Book depends upon the eighth of the fame; and this eighth, and the Corollary of Propofition 17th and Propofition 18th of the 12th Book, depend upon the 9th Definition of the 11th Book, which is not a right definition; because there may be folids contained by the fame number of fimilar plane figures, which are not fimilar to one another, in the true fenfe of fimilarity received by geometers; and all thefe Propofitions have, for these reafons, been infufficiently demonftrated fince Theon's time hitherto. Befides, there are feveral other things, which have nothing of Euclid's accuracy, and which plainly fhew, that his Elements have been much corrupted by unfkilful geometers; and, though thefe are not fo grofs as the others now mentioned, they ought by no means to remain uncorrected.
Upon these accounts it appeared neceffary, and I hope will prove acceptable, to all lovers of accurate reafoning, and of mathematical learning, to remove fuch blemishes, and reftore
the principal Books of the Elements to their original accuracy, as far as I was able; especially fince thefe Elements are the foundation of a fcience by which the investigation and difcovery of useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewife is of the greateft ufe in the arts both of peace and war, to many of which geometry is abfolutely neceffary. This I have endeavoured to do, by taking away the inaccurate and falfe reafonings which unfkilful editors have put into the place of fome of the genuine Demonftrations of Euclid, who has ever been justly celebrated as the most accurate of geometers, and by restoring to him thofe things which Theon or others have fuppreffed, and which have these many ages been buried in oblivion.
In this twelfth edition, Ptolemy's Propofition concerning a property of quadrilateral figures in a circle, is added at the end of the fixth Book. Alfo the Note on the 29th Propofition, Book Ift, is altered, and made more explicit, and a more general Demonftration is given, instead of that which was in the Note on the 10th Definition of Book 11th; befides, the Tranflation is much amended by the friendly affiftance of a learned gentleman.
To which are alfo added, the Elements of Plane and Spherical Trigonometry, which are commonly taught after the Elements of Euclid.
Point is that which hath no parts, or which hath no magnitude. See Notes.
A line is length without breadth.
The extremities of a line are points.
A straight line is that which lies evenly between its extreme points.
A fuperficies is that which hath only length and breadth.
The extremities of a fuperficies are lines.
A plane fuperficies is that in which any two points being taken, See N. the straight line between them lies wholly in that fuperficies.
"A plane angle is the inclination of two lines to one another in a See N. "plane, which meet together, but are not in the fame direction."
A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the fame ftraight line.
• N.B. When several angles are at one point B, any one of them is expreffed 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 ftraight lines AB, CB is named the angle ABC, or CBA; that which is contained by AB, DB is named the angle ABD, or DBA; and that which is contained by DB, CB is called the angle DBC, or CBD. but ❝ if there be only one angle at a point, it may be expreffed by a • letter placed at that point; as the angle at E.'
When a ftraight line standing on another
An obtufe angle is that which is greater than a right angle.
An acute angle is that which is lefs than a right angle.
"A term or boundary, is the extremity of any thing."
A figure is that which is inclosed by one or more boundaries