The Doctrine of the Dimension of PLAIN and SPHERICAL TRIANGLES is Succinctly Handled, Geometrically Demonstrated,

By William Leybourn, Philomathemat.

LONDON:

Printed Anno Domini, MDCCIV.

JANUA MATHEMATICA.

SECTION I.

Of Plain (or Right Lined) Triangles.

CHAP. I..

DEFINITIONS.

F Triangles there are Two Kinds; viz. Plain, (or Rightlined) and Spherical, (or Circular.) Either of which do confift of Six Parts; namely, of Three Sides, and as many Angles; but in this Place we shall only treat of the Plain.

I. A Plain (or Right Lined) Triangle consisteth of Three Sides, Fig. I. and as many Angles: And such are the Two Figures CBA and CDB; in which, in the first Figure AB, BC and CA, are the Three Sides of the Triangle CBA; and CA B, ACB and CBA, are the Three Angles of the same Triangle C B A.

And Note here] That an Angle (in any Cafe) is always noted with Three Letters, the middlemost whereof repreSents the Angular Point. So in the Triangle ABC, if I would express the Angle at C, I would fay, The Angle ACB; or, The Angle BC A.

Also in the second Figure D B C, the Lines DC, D B and CB, are the Three Sides of the Triangle DBC; and the Angles BDC, CBD. and B CD, are the Angles of the same Triangle DBC.

Fig. 1.

Fig. Π.

II. Any Two Sides of a Triangle, are called the Sides of that Angle contained by (or comprehended between) them: So the Sides BC and CA are the Sides containing the Angle BC A.

III. Every Side of a Triangle, is the subtending Side of that Angle which is opposite unto it. As in the Triangle ABC, BC is the fubtending Side of the Angle CAB; the Side B A fubtends the Angle BCA, and the Side C A fubtends the Angle СВА.

And here Note again,] That the greatest Side, always, Subtends the greatest Angle, the lesser Side the leffer Angle, and equal Sides fubtend equal Angles.

IV. The Measure of an Angle is the Arch of a Circle described upon the Angular Point, and is intercepted between the Two Sides containing the Angle, (increasing the Sides, if Need be). So in the Triangle ABC, the Measure of the Angle CBA, is Arch

CF.

V. Every Circle is divided into 360 Degrees, and every Degree into 60 Minutes, (or rather into 100 or 1000 Parts, &c.) Which Degrees are so much the Greater, by how much the Circle is Greater; and those Arches which contain the fame Number of Degrees in equal Circles, are Equal: But in unequal Circles they are termed Like-Arches. So the Arches CF and DE are Equal Arches, they being equal Parts of the fame Circle DHFK. But the Arches CF and OP are Like-Arches: For, as CF is 40 Parts (or Degrees) of the Greater Circle DEFC, fois OP 40 Degrees of the Leffer Circle POG.

VI. A Quadrant (or Quarter) of a Circle, is an Arch of 90 Degrees. As is the Quadrant (or Arch) H F.

VII. The Complement of an Arch less than a Quadrant, is s much as that Arch wanteib of 90 Degrees. So the Complement o the Arch CF 40 Degrees, is the Arch HC 50 Degrees.

as

VIII. The Excess of an Arch Greater than a Quadrant, is so ma ny Degrees that Arch exceedeth 90 Degrees. So the Arc DHC being 140 Degrees, is the Arch H C, 50 Degrees mor than the Quadrant DH.

IX. A Semicircle is an Arch of 180 Degrees. As is the Semicil cle DH F.

X. The Complement of an Arch less than a Semicircle, to a S micircle, is so much as that Arch wants of 180 Degrees. So t Complement of the Arch DHC 140 Degrees, to the Semicir DHE 180 Degrees, is the Arch CF 40 Degrees.

XI

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XI. The oppofite Angles made by the croffing of Two Diameters Fig. II.

in a Circle, (or any Two other Right Lines croffing each other) are equal. So the Angles CBF and CB E, (made by the Intersection of the Two Diameters DF and CE in the Center B) are equal.

XII. An Angle is either Right or Oblique.

XIII. A Right Angle is that whose Measure is a Quadrant or 90 Deg. So the Angles HBD, and H B F, are Right Angles, their Measures being the Quadrants DH and HF.

XIV. All Oblique Angles are either Acute or Obtuse.

XV. An Acute Angle, is that whose Measure is less than 90 Deg. So the Angles HBC 50 Deg. and CBF 40, are Oblique Acute Angles.

XVI. An Obtufe Angle, is that whose Measure is more than a Quadrant or 90 Deg. So the Angle DBC (confifting of 90 and 50 Deg. viz. of 140 Deg) is an Oblique Obtufe Angle.

XVII. The Complements of Angles are the same, as are the Complements of Arches.

XVIII. All Angles concurring (or meeting) together upon One Right Line, all of them being taken together, are equal to a Semicircle, or 180 Deg. So the Angle DBH 90 Deg. HBC 50 Deg. and CBF 40 Deg. (made by the concurring, or meeting, of the Three Lines DB, HB and CB, upon the Diameter DF, in the Center B) are all of them Equal to the Semicircle DHCF, or 180 Deg.

XIX. A Triangle hath some of its Sides Equal, or else they be all Unequal.

XX. A Triangle of some Equal Sides is either Equicrural or Equilateral.

XXI. An Equicrural Triangle is that which hath only Two Equal Sides. And fuch is the Triangle DBE, whose Sides BD and BE are Equal.

XXII. An Equicrural Triangle is Equi-angled at the Base. So in the Equicrural Triangle BDE, the Angles BD E and BED, at the Base DE, are Equal, viz. each of them 70 Deg. for the Angle DBE being 40 Deg. that taken from 180 Deg. leaves140 Deg. the half, whereof 70 Deg. is equal to the Angle BDE or DEB, and all the Three Angles equal to 180 Deg. or Two Right Angles.

This is Demonstrated in the XIth Theorem hereof,

XXIII. An