5th Book, by which the doctrine of compound Ratios is ren
dered plain and easy. Befides, 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 equa-
lity of Figures of any kind must be demonftrated, and not affu-
med; and, therefore, though this were a true Propofition, it
ought to have been demonftrated. But, indeed, this Propo-
fition, 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 Magni-
tude, and yet be unequal to one another, as fhall be made evi-
dent in the Notes fubjoined to thefe elements. In like manner,
in the Demonstration of the 26th prop. of the 11th Book, it is
taken for granted, that thofe folid 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 Demonstration in the Elements we
now have, though it be quite neceffary there should be one.
Now, upon the 10th 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 17. and Prop. 18th 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 fame number of fimilar plane Figures, which
are not fimilar to one another, in the true Sense of Similarity
received by Geometers; and all thefe Propofitions have, for
thefe Reasons, 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 Geome-
ters; and, though these are not fo grofs as the others now
mentioned, they ought by no means to remain uncorrected.

Upon thefe Accounts it appeared neceffary, and I hope will
prove acceptable to all Lovers of accurate Reasoning, and of
Mathematical Learning, to remove fuch Blemishes, and restore

the

the principal Books of the Elements to their original Accuracy, as far as I was able; especially fince 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 likewife is of the greatest Use 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 Reasonings which unskilful Editors have put into the place of fome of the genuine Demonftrations of Euclid, who has ever been juftly 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 Ninth 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 Prop. Book ift, 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 11th; befides, the Tranflation is much amended by the friendly affiftance 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.

THE

ELEMENTS

EUCLI D.

воок ь.

DEFINITIONS.

I.

A

Point is that which hath no parts, or which hath no See Notes. magnitude.

II.

A line is length without breadth.

III.

The extremities of a line are points.

IV.

A ftraight line is that which lies evenly between its extreme points.

V.

A fuperficies is that which hath only length and breadth.
VI.

The extremities of a fuperficies are lines.

VII.

A plane fuperficies is that in which any two points being taken, the straight line between them lies wholly in that fuperficies. VIII.

See N.

"A plane angle is the inclination of two lines to one another "in a plane, which meet together, but are not in the fame See N. "direction."

IX.

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.

Book I. (y)

C

N. B. When feveral 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 ftraight lines that contain the angle meet one another, is put between the other two letters, and one of thefe two is fomewhere 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, 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 expreffed by a letter placed at that point; as the Angle at E.'

X.
When a ftraight line ftanding on ano-
ther ftraight line makes the adjacent
angles equal to one another, each of
the angles is called a right angle;
and the ftraight line which ftands on
the other is called a perpendicular to

it.

XI.

An obtufe angle is that which is greater than a right angle.

XII.

An acute angle is that which is less than a right angle.
XIII.

"A term or boundary is the extremity of any thing."
XIV.
A figure is that which is inclefed by one or more boundaries,