AXIOME I. In a Right-angled Plain Triangle: The Rectangle made of Radius, and one of the Sides containing the Right Angle, is equal to the Rectangle, made of the other containing Side, and the Tangent of the Angie thereunto adjacent. DEMONSTRATION. In the Right-angled Plain Triangle B E D, draw the Arch F E; Fig. VIII. then is B E Radius, and D the Tangent of the Angle at B: Make CA parallel to DE, then are the Triangles A B C and B DE like, because of their Right Angles at A and E, and their common Angle at B. Therefore, As BABE:: AC: ED. And, B A in ED is equal to B E in A C. That is, BA in t B is equal to Radius in A C. Which which was to be demonftrated. AXIOME II. In all Plain Triangles: The Sides are proportional to the Sines of their oppofite angles. DEMONSTRATION. In the Plain Oblique Triangle C B D, extend BC to F, making Fig. IX. BF equal to DC, and defcribe the Arches F G and CH, then are the Perpendiculars F E and C A, the Sines of the Angles at D and B. and the Triangles B E F and B A C are like, because of their Right Angles at E and A, and their common Angle at B. Therefore, Fig. X. AXIO ME III. In all Plain Triangles: As the balf Sum of the Sides, is to their half Difference, fo is the Tangent of the half Sum of their oppofite Angles, to the Tangent of their half Difference. Otherwise, In every Plain Triangle: As the Sum of the Two Sides, is to their DEMONSTRATION. In the Oblique angular Triangle A B C, let the known Sides be B A and BC; and the Angle ABC, comprehended by them: Where it is Obtufe in the fuperior, but Acute in the inferior, Diagram. Continue the Side A B to H; fo that H B may be equal to BC, and join C and H, -and make BI equal to A B: Alfo, from the Points B and I, draw the Right Lines BD and IG, parallel to the Side A C. Then shall the exterior Angle C B H be equal to the Two interior and oppofite Angles (by the 32th of the 1ft Euclid.) For the Angle CBD is equal to the Angle A CB, and the Angle DB H, to the Angle C A B. (Moreover, from the Point B, let fall a. Perpendicular B E, which fhall bifect C H at the Point E; then making B E the Radius, upon B, defcribe the Arch ME L. Therefore shall CE be the Tangent of half the Sum of the oppofite Angles: And DE (to which FE is equal) the Tangent of half their Difference. Now, because AC, BD, and IG, are Parallel, therefore shall CD, DG, FH, be Equal: As alfo, D F and GH: Therefore I fay, By Equality of Proportion: As A H, the Sum of the Two Sides, Is to I H, their Difference; So is CE, the Tangent of half the oppofite Angles, Otherwife, As the Sum of the Two Sides, Is to the greater Side doubled; So is the Tangent of half the Sum of the oppofite Angles, To the Sum of the Tangents of the half Sum, and half Difference of the Angles. Otherwise, As the Sum of the Two Sides, Is to the leffer Side doubled; So is the Tangent of half the Sum of the oppofite Angles, AXIOME IV: In all Plain Triangles: As the Bafe, is to the Sum of the other DEMONSTRATION. In the Triangle BCD, let fall the Perpendicular CA; extend Fig. XI. B C to F, and draw F G and D H. Then is B F equal to the Sum of the Sides CD and CB; and HB equal to the Difference of thofe Sides; and GB is equal to the Difference between A B and AD, the Segments of D B, the Bafe: And the Triangles HDB and B G F are like; because of their equal Angles at D and F; the Arch H G being the double Measure of them both: And their common Angle at B. Therefore, As B D BF: HB: G B. That is, As DB:FB (the Sum of the Sides): H B (the Difference of the Sides C B and CD): GB (the Difference of the Segments of the Bafe.) ADVERTISEMENT. this Part A Table to Reduce Sexagenary Mi-Pereas in book nutes, to Centefimal Parts; and 08.33 6 .10. 71.11.67 8.13.33 Cent. Parts. 31.51.67 32.53.33 33.55. 34.56.67 35 58.33 26 .60. 37 .6167 38.63.33 51.85. 54.90. W of this which treateth of Trigonometry, where the Angles of Right-lined, and the Sides and Angles of Spherical Triangles are meafured by Arches of Circles; and thofe Arches are ufually numbred (or accounted) by Degrees, Minutes, Seconds, &c. Of which one Degree contains 60 Minutes, One Minute 60 Seconds, &c. which is called the Sexary Divifion of the Degree. Ör otherwife, the Degree is fuppofed to be divided into 100 or 1000 Parts, which is called the Cen tefimal, Millefimal, &c. Divifion, (and is the better of the Two in many Respects.) Now, whereas in the following Trigonometrical Calculations of this Book, and alfo in the other Parts (which concern the Doctrine of Triangles applied to Practice in feveral Parts of the Mathematicks) I have, fometimes, ufed the Sexagena ry, and fometimes the Centefimal, Way of Di |