Writers of Trigonometry, that I have met with, who have undertaken the Solutions of these two, as well as the two following Cafes, by letting fall a Perpendicular, which is undoubtedly the shortest and best Method for finding either of thefe Quæfita,

}

which can

Vertical Angles, or Bafes, fhall be the fought Angle or Side, according as the

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without;

Perpendicular falls within;

not be known, unless the Species of that unknown Angle, which is oppofite to a given Side, be first known.

Here they leave us firft to calculate that unknown Angle, before we shall know whether we are to take the Sum or the Difference of the vertical Angles or Bafes for the fought Angle or Bafe: And in the Calculation of that Angle have left us in the dark as to its Species; as appears by the Obfervations on the two preceding Cafes.

The Truth is, the Quafitum here, as well as in the two former Cafes, is fometimes doubtful, and fometimes not; when doubtful, fometimes each Answer is lefs than 90 Degrees, fometimes each is greater; but fometimes one lefs, and the other greater, as in the two laft-mentioned Cafes. When it is not doubtful, the Qua

fitum is fometimes greater than 90 Degrees, and fometimes lefs; all which Diftinctions may be made without another Operation, on 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.

Sum

In the Solution of our 1ft and 5th Cases, 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 Bases, according within, as the Perpendicular falls without, fhall 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 Quafitum 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 of 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 laft-mentioned, is fometimes ambiguous, and fometimes 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, fee in Page

224.

SAM. CUNN.

THE

Advertisement.

HE Reader is now presented with a more correct Edition of this Work, than any hitherto extant; for not only many Typographical Erorrs had by Degrees crept into it, but there were many Omiffions and Miftakes, even in the firft Edition, the greater Part of which have been conftantly adhered to, in the five subsequent 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 Master of the Royal Mathematical School in Ghrift's Hofpital, with an interleaved Copy of the firft Edition thereof, in which are a great Number of Additions and Corrections of Mr. Cunn's own hand-writing, defigned (as may be fupposed) 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, to this Edition, and several 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 fixth Edition, the Revifor carefully perused the fame, and rectified great Numbers of falfe References to the Plates, and fome Errors in the Plates themselves (for they are not the fame with those annexed to the First Edition): But the most Aagrant 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 several 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 evenly between its Lines.

VIII. A plane Angle is the Inclination of twa 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