fitum is fometimes greater than 90 Degrees, and fometimes lefs; all which Diftinctions may be made without another Operation, or the Knowledge of the Species of that unknown Angle, oppofite to a given Side; or, which is the fame Thing, the Falling of the Perpendicular within or without. For which, fee Pages 323, 324.

In the Solution of our 1ft and 5th Cafes, called in other Authors the 5th and 6th; where there are given two Angles, and a Side oppofite to one of them, to find the third Angle, or the Side oppofite to it;

they have told us, that the Difference }

of the vertical Angles, or Bafes, according as the Perpendicular falls within,

without, hall be the fought Angle or Side; and that it is known whether the Perpendicular falls within, or without, by the Affection of the given Angles.

Here they seem to have spoken as tho' the Quæfitum was always determined, and never ambiguous; for they have here determined whether the Perpendicular falls within or without, and thereby whether they are to take the Sum or the Difference of the vertical Angles or Bafes for the fought Angle or Side.

But,

But, notwithstanding these imaginary Determinations, I affirm, that the Quafitum here, as in the two Cafes last-mentioned, is fometimes ambiguous, and fometimes not; and that too, whether the Perpendicular falls within, or whether it falls without. See the Solutions of thefe two Cafes in Page 322.

The Determination of the 3d Cafe of Oblique Plane Triangles, fee in Page 224.

SAM. CUNN.

TH HE Reader is now prefented with a more correct Edition of this Work, than any hitherto extant; for, not only many Typographical Errors had by Degrees crept into it, but there were many Omiffions and Mistakes, even in the First Edition, the greater Part of which have been conftantly adhered to, in the five fubfequent ones. Upon the Application of the Proprietors for a Revifion of this Work, the Revifor was favoured, by Mr. John Robertfon, F.R. S. late Mafter of the Royal Mathematical School in Christ's Hofpital, with an interleaved Copy of the first Edition thereof, in which are a great Number of Additions and Corrections of Mr. Cann's own Hand-writing, defigned (as may be fuppofed) to have been inferted in a Second Edition; but probably, prevented from fo being, either by his Death, or fome other Accident: All thefe Alterations have been carefully made, in this Edition, and feveral more Errors, even in that Edition which had escaped Mr. Cunn's Notice, and have been continued in the following Editions, are in this corrected.

After these Amendments had been made, in the printed Copy of the Sixth Edition, the Revifor carefully perufed the fame, and rectified great Numbers of false References to the Plates, and fome Errors in the Plates themselves (for they are not the fame with thofe annexed to the First Edition): But the most flagrant Typographical Errors appeared in the Algebraic Series, given in the Treatifes on Trigonometry and Logarithms, and demonftrated in the Appendix; for the greatest Part of these were fo badly difpofed, as to be unintelligible, even to those who understand the Subject; these are here rendered intelligible, and the Whole now is (as the Revifor apprehends) in fuch a State, as the feveral Authors of the Work and Appendix would have chose to have put it into, had they been alive fo to do.

EUCLID's

ELEMENTS.

BOOK I.

DEFINITION S.

1. A POINT is that which hath no Parts

or Magnitude.

II. A Line is Length, without Breadth.

III. The Ends (or Bounds) of a Line are Points.

IV. A Right Line is that which lieth evenly between its Points.

V. A Superficies is that which hath only Length and Breadth.

VI. The Bounds of a Superficies are Lines. VII. A plane Superficies is that which lieth even ly between its Lines.

VIII. A plane Angle is the Inclination of two Lines to one another in the fame Plane, which touch each other, but do not both lie in the fame Right Line.

IX. If the Lines containing the Angle be Right ones, then the Angle is called a Right-lined Angle.

X. When one Right Line, ftanding on another Right Line, makes Angles on either Side there

B

of,

of, equal between themselves, each of these equal Angles is a Right one; and that Right Line, which stands upon the other, is called a Perpendicular to that whereon it ftands.

XI. An Obtufe Angle is that which is greater than a Right one.

XII. An Acute Angle is that which is less than a Right one.

XIII. A Term (or Bound) is that which is the Extreme of any Thing.

XIV. A Figure is that which is contained under one or more Terms.

XV. A Circle is a plain Figure, contained under one Line, called the Circumference; to which all Right Lines, drawn from a certain Point within the Figure, are equal.

XVI. And that Point is called the Centre of the Circle.

XVII. A Diameter of a Circle is a Right Lino drawn through the Centre, and terminated on both Sides by the Circumference, and divides the Circle into two equal Parts.

XVIII. A Semicircle is a Figure contained under a Diameter, and that Part of the Circumference of a Circle cut off by that Diameter. XIX. A Segment of a Circle is a Figure contained under a Right Line, and Part of the Circumference of the Circle [which is cut off by that Right Line.]

XX. Right-lined Figures are fuch as are con tained under Right Lines.

XXI. Three-fided Figures are fuch as are contained under three Lines.

XXII. Four-fided Figures are fuch as are contained under four Lines.

XXIII. Many-fided Figures are those that are contained under more than four Right Lines.