JANUA MATHEMATICA. SECTION I Of Plain (or Right Lined) Triangles. CHA P. I.. DEFINITION S. Ο 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. fhall only treat of the Plain. I. A Plain (or Right Lined) Triangle confifteth of Three Sides, Fig. I. and as many Angles: And fuch are the Two Figures CBA and CDB; in which, in the firft Figure A B, BC and C A, are the Three Sides of the Triangle CBA, and CA B,. A CB and CBA, are the Three Angles of the fame Triang CB A. And Note here] That an Angle (in any Cafe) is always noted with Three Letters; the middlemoft whereof reprefents the Angular Point. So in the Triangle ABC, if I would exprefs the Angle at C, I would fay, The Angle. ACB; or, The Angle BC A. Alfo in the fecond Figure D B C, the Lines DC, D B and CB, are the Three Sides of the Triangle DB C, and the Angles BDC, CBD and B C D, are the Angles of the fame Triangle DBC. II. Any Fig. 1. Fig. II. II. Any Two Sides of a Triangle, are called the Sides of that Angle contained by (or comprehended between) them: So the Sides B C and CA are the Sides containing the Angle BC A. III. Every Side of a Triangle, is the fubtending Side of that Angle which is oppofite 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 CBA. And here Note again,] That the greateft Side, always, Subtends the greatest Angle, the leffer Side the leffer Angle, and equal Sides fubtend equal Angles. IV. The Meafure of an Angle is the Arch of a Circle defcribed upon the Angular Point, and is intercepted between the Two Sides containing the Angle, (increafing the Sides, if Need be). So in the Triangle A B C, the Measure of the Angle CBA, is Arch C F. 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 fo much the Greater, by how much the Circle is Greater; and thofe 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 C F and DE are Equal Arches, they being equal Parts of the fame Circle DHFK. But the Arches CF and O'P are Like-Arches: For, as CF is 40 Parts (or Degrees) of the Greater Circle DE FC, 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 fo much as that Arch wanteth of 90 Degrees. So the Complement of the Arch C F 40 Degrees, is the Arch HC 50 Degrees. VIII. The Excefs of an Arch Greater than a Quadrant, is fo many Degrees as that Arch exceedeth 90 Degrees. So the Arch DHC being 140 Degrees, is the Arch H C, 50 Degrees more than the Quadrant DH. IX. A Semicircle is an Arch of 180 Degrees. As is the Semicirele DHF. X. The Complement of an Arch less than a Semicircle, to a Semicircle, is fo much as that Arch wants of 180 Degrees. So the Complement of the Arch D HC 140 Degrees, to the Semicircle D HE 180 Degrees, is the Arch C F 40 Degrees. XI. The 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 C B F and CB E, (made by the Interfection 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 whofe Measure is a Quadrant or 90 Deg. So the Angles H BD, and HBF, are Right Angles, their Measures being the Quadrants DH and H F. XIV. All Oblique Angles are either Acute or Obtufe. XV. An Acute Angle, is that whofe Meafure is lefs than 90 Deg. So the Angles HBC 50 Deg. and C B F 40, are Oblique Acute Angles. XVI. An Obtufe Angle, is that whofe Meafure is more than a Quadrant or 90 Deg. So the Angle DB C (confifting of 90 and 50 Deg. viz. of 140 Deg ) is an Oblique Obtufe Angle. XVII. The Complements of Angles are the fame, 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, H B and C B, upon the Diameter DF in the Center B) are all of them Equal to the Semicircle D HCF, or 180 Deg. XIX. A Triangle hath fome of its Sides Equal, or elfe they be all Unequal. XX. A Triangle of fome Equal Sides is either Equicrural or Equilateral. XXI. An Equicrural Triangle is that which hath only Two Equal Sides. And fuch is the Triangle D B E, whofe Sides BD and BE are Equal. XXII. An Equicrural Triangle is Equi-angled at the Bafe. So in the Equicrural Triangle BDE, the Angles B D E and BED, at the Bafe D E, 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 Demonftrated in the XIth Theorem hereof. XXIII, An Fig. II. XXIII. An Equilateral Triangle, is that whofe Sides are all Equal, and whofe Angles contain (each of them) 60 Deg. So the Triangle E BK hath its Sides BE, B K and E K, all of them equal; and the Angles E BK, EK B and KE B equal alfo, and each of them equal to 60 Deg. and confequently all of them equal to 180 Deg. XXIV. A Triangle is either Right Angled or Oblique Angled. XXV. A Right-angled Triangle, is that which hath one Right Angle. And fuch is the Triangle C A B, Right-angled at A. XXVI. An Oblique-angled Triangle is that which hath all its Angles Oblique. And fuch is the Triangle B CD. XXVII. An Oblique-angled Triangle, is either Acute-angled, or Obtufe-angled. XXVIII. An Oblique Acute-angled Triangle is that which hath all its Three Angles Acute. And fuch are the Triangles D B E, and EB K. XXIX. An Oblique Obtufe-angled Triangle is that which hath One Obtufe, and Two Acute Angles. And fuch is the Triangle DBC, whofe Angle DBC is Obtufe, and the Angles BDC and BCD Acute. Fig. III. CHA P. II. Of Right Lines, applied to a Circle. Orafmuch as the Ratio or Proportion of an Arch Line to a Right Line, is as yet unknown, yet it is abfolutely neceffary that Right Lines be applied to a Circle, for the Calculation of Triangles, wherein Arch Lines come in Competition: For the Angles of Plain (or Right-lined) Triangles are measured by Arches of Circles. Now, the Right Lines applied (or relating) to a Circle, are 1. A Chord, or Subtenfe, is a Right Line, joining the Extremities of an Ark, as A C is the Chord of the Arks ABC and ADC. 2. A Right Sine, which is fingly called a Sine, is a Right Line, drawn from one end of an Ark, perpendicular to the Diameter drawn through to the other End: Or, it is half the Chord of twice the Ark; fo AE is the Right Sine of the Arks A B and A D. The Radius (or Sine of 90 Deg.) is called the Whole Sine, and is the greatest of all Sines: For the Sine of an Ark Fig. III. greater than a Quadrant, is lefs than the Radius; fo F G is the whole Sine or Radius. 3. A Verfed Sine is the Segment of the Radius between the Ark and its Right Sine; fo E B is the Verfed Sine of the Ark A B, and of the Ark AG D. 4. The Secant of an Ark, is a Right Line, drawn from the Center through one end of an Ark, till it meet with the Tangent: That is, a Right Line touching the Circle at the neareft end of that Diameter which cuts the other end of the Ark. FM is the Secant, and BM the Tangent, of the Ark A B, or of A D. 5. The Difference of an Ark from a Quadrant, (or 90 Deg.) whether it be Greater or Lefs, is called the Complement of that Ark, foGA is the Complement of the Arks A B and A G D, and HA is the Sine of that Complement: GI the Tangent of that Complement and F I the Secant of that Complement. All which (for Brevity) we write Co-Sine, Co-Tangent, Co-Secant of the Ark. 6. The Difference of an Ark from a Semicircle (or 180 Deg.) is called its Supplement; fo the Ark A B is the Supplement of the Ark DG A, to a Semicircle. 7. That Part of the Radius which is between the Centre and its Right Sine, is equal to the Co-Sine. As FE is equal to H A, and FO is equal to the Co-Sine of the Ark DS. 8. If an Ark be Greater or Lefs than a Quadrant, the Sum or Difference, accordingly, of the Radius and Co-Sine, is equal to the Verfed Sine For F D and H A together, are equal to D E, the Verfed Sine of the Ark DGA; and F B lefs by HA (or E F) is equal to E B, which is the Verfed Sine of the Ark A B. |