fitum is sometimes greater than 90 Degrees, and sometimes less; all which Distinctions may be made without another Operation, or the Knowledge of the Species of that unknown Angle, opposite to a given Side ; or, which is the same Thing, the Falling of the Perpendicular within or without. For which, see Pages 323, 324. 3 } In the Solution of our ist and 5th Cafes, called in other Authors the 5th and 6th where there are given two Angles, and a Side opposite to one of them, to find the third Angle, or the Side opposite to it; they have told us, that the Difference of the vertical Angles, or Bases, according as the Perpendicular falls within, shall 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. withour,}hali Here they seem to have spoken as tho' the Quchtum 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 Bases for the sought Angle or Side. But, But, notwithstanding these imaginary Determinations, I affirm, that the Quæfitum here, as in the two Cafes last-mentioned, is sometimes ambiguous, and sometimes not; and that too, whether the Perpendicular falls within, or whether it falls without. See the Solutions of these two Cafes in Page 322. The Determination of the 3d Cafe of Oblique Plane Triangles, see in Page 224 THI HE Reader is now presented with a more cor rect Edition of this Work, than any hitherto extant; for, not only many Typographical Errors had by Degrees crept into it, but there were many Omissions and Mistakes, even in the First Edition, the greater Part of which have been constantly adhered to, in the five subsequent ones. Upon the Application of the Proprietors for a Revision of this Work, the Revisor was favoured, by Mr. John Robertson, F.R. S. late Mafter of the Royal Mathematical School in Chrill's Hospital, with an interleaved Copy of the first Edition thereof, in which are a great Number of Additions and Corrections of Mr. Cunn's own Hand-writing, designed (as may be supposed) to have been inserted in a Second Edition; but probably, prevented from so being, either by his Death, or some other Accident: All these Alterations have been carefully made, in this Edition, and several more Errors, even in that Edition which bad escaped Mr. Gunn'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 Revisor carefully perused the same, and rectified great Numbers of false References to the Plates, and some Errors in the Plates themselves (for they are not the same with those annexed to the First Edition): But the most flagrant Typographical Errors appeared in the Algebraic Series, given in the Treatises on Trigonometry and Logarithms, and demonstrated in the Appendix; for the greatest Part of these were so badly disposed, as to be unintelligible, even to those who understand the Subject; these are here rendered intelligible, and the Whole now is (as the Revisor apprehends) in such a State, as the several Authors of the Work and Appendix would have chose to have put it into, had they been alive fo to do. EUCLID's E LE MEN T S. BOOK I. DEFINITIONS. 1. A POINT is that which bath no Parts or Magnitude. II. A Line is Length, without Breadtb. III. The Ends (or Bounds) of a Line are Points. IV. A Right Line is that which lieth evenly be tween its Points. V. A Superficies is that which hath only Length and Breadıb. VI. The Bounds of a Superficies are Lines. VII. A plane Superficies is that which lietb even. ly between its Lines. VIII. A plane Angle is the Inclination of two Lines to one another in the same Plane, which touch each other, but do not boib lie in the Same 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, standing on another Right Line, makes Angles on either Side thereB 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 Per pendicular to that wbereon it stands. XI. An Obtuse 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 thai 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 Paris. XVIII. A Semicircle is a Figure contained under a Diameter, and that part of the Circumfe rence 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 such as are con tained under Right Lines. XXI. Three-fided Figures are such as are con tained under three Lines. XXII. Four-sided Figures are such as are con tained under four Lines. XXIII. Many-sided Figures are those that are contained under more than four Right Lines. |