Part of the Middle Term, is called the Middle Part, and the Fig.XXI. V. But the Extream Parts may be twofold, either Conjunt, or VI. But of those Three Terms which may fall in question, we will fubje&t all their Varieties in their Circular Parts, according as every one of them ought (in respect of each other) to be called, The Middle Part; and which, The Extreams Conjunct or Disjunct, as in this following Synopsis is fully demontrated. If the Middle Part. Side Side Compl. Compl. Compl. B C Side AC and Comp. B | Com.BCand Com.C Side AB and Com. B Comp. B and Comp. C | Side AB & Side A C. Which Synopsis is thus to be Read and Understood: Example of the First Line. If the Side A B be the Middle Part, then is the Side A C, and Comp. of the Angle B, the Extreams Conjunct. And the Comp. of the Side B C. and Comp. of the Angle C, the Extreams Disjunct. And fo of all the reft. And here it is to be noted, That the Sides AB and A C, are fupposed to be joined together, (as one entire Part,) because the Right Angle at A, is not reckoned amongst the Circular Parts. VII. Therefore, In the Resolution of a Right-angled Spherical Triangle, to know the Mean, and Extream Parts, you must observe, That 1. If One of the Three Terms (which, befides the Right Angle, come in question) doth stand alone by it felf, severed from the i Fig.XXI. the other Two on both Sides; (as the Side B C, from the Sides CA and B A, by the Angles B and C interposed) that shall be the Middle Term; and so its Circular Part shall be called the Middle Part; and the other Two Circular Parts are the Extreams Disjunct. But, 2. If the Three Terms do immediately adhere together, the Middle Term doth easily shew the Middle Part, and the Extream Terms, the Extream Parts Conjunct. These Things being all rightly understood, the whole Trigonometry of Sphericals will be abfolved by this One Propofition; which therefore we will call Catholick or Univerfal. Proposition Universal. The Sine of the Middle Part and the Radius, are Reciprocally Proportional, with the Tangents of the Extream Parts Conjunct; and with the Co-fines (or Sines Complements) of the Extreams Disjunct. That is, As the Radius, To the Tangent of one of the Extreams Conjunct; As the Radius, Then also, To the Co-fine of one of the Extreams Disjunct; COROLLARY I. If the Middle Part be Sought, the Radius shall be in the First Place of the Proportion: But if one of the Extream Parts be Sought, then the other Extream shall be in the First Place. Of the Second and Third Places, it mattereth nothing how they be difpofed. II. If II. If the Extreams (in any Proportion) be Distinct from the Mid. Fig.XXI. dle Part, the Proportion will be performed by Sines only: But if the Extreams be Conjunt to the Middle Part, it must be performed by Sines and Tangents jointly. The Demonstration of the Universal Propofition is obvious enough: For, where the Extream Parts are Disjunct, the Proportions differ nothing from the common ones: And in the Extreams Conjunct, where it is commonly said, As Radius, to the Tangent, We here fay, As the Co-Tangent, to the Radius. And likewise Inversly and Contrarily; which is plainly the fame The Radius is a Mean Proportional, between the Tangent of Note, That when a Complement in any Proportion doth chance to As is the Sixth Cafe following; where CB and C A are given, These Things premised, we will exemplifie in the Solution of CHAP |