DEMONSTATION. If HG and FI be drawn through any point E of the Diagonal CB parallel to AC and CD, the Parallelogram ACDB is divided into our Parallelograms, two of which are about the Diagonal, and the other two are their Coinplements, which are thus hewn to be equal, ABC to a CDB and the as HBE and EFC are to as IBE and ECG (by 8) If from the Equal As ABC and CDB you subftra& the Equal As HBE, EFC and IBE, CEG there will remain the Parall-logram AHEF to the Parallelogram EGID (by 2d Axiom.) THEOREM XII. Fig. 52. In every rectangle Triangle A B C, the Square of the fide AC, which is oppofite to the right Angle, is equal to the Squares of the other two fides (A B, CB) Demon. Draw IC,BF,and BE,Parallel to AF. I then you add the common < BAC to the right Angles IAB, FAC, and therefore equal, the wholes IAC, FAB, will be equal, but the 4s IAC, FAB have the two fides which contains thofe Angles equal (by Def.15) to wit IA BA and CA FA.. ◄ IAC ▲ FAB (by Prop. 1) but a IAC = ÷ the Square ILBA and ▲ ABF Parallelo gram AFZ the (by Prop.10) therefore Square: LIBA =the Parallelogram AFZE. I might te fhewn with the fame Eafe that the Square BXCH the Parallelogram CZER. Q. E D. THEOREM XIII: Fig.53. An <(BCA) at the Center is double to the < (AFB) at the Circumference when the fame Arc (AB) is Bife to both Angles. This Prop. bath three Cafes, The first is when the fide (CA) coincides with the fide (AF). For then CF CB, because both ara drawn from the Center to the Circumference of the fame Circle therefore in a CFB<CBF =< CFB (per Prop. 2) but <BCA =< CBFCFB (per Schol. 7 Prop.). < ACB is double the <CFB which may the firft In the fecond Cafe CA and CB tall without AF and BF. Then <XCA is double < A FX, and < XCB is double the <XFB (by the fift Cafe) Theretore the whole < ACB is donble the whole < AEB. In the third Cafe RK cuts C A, and the < AKB is wholly without the < ACB Draw KCL then ACL is double < AKL (by the first Cafe) and if LCB and its double < LKB be taken away there remains <A ́B doub.e< AKB. Q. E. D. THEOREM. XIV. All fimilar Triangles have their fides about their equal Angles proportional. For if they were infcribed in Circles, their Șides would be Chords of fimilar Arcs. THEOREM. XV. Fg. 54. If in any Triangle a Line be drawn parallel to the Bafe, that Line will cut the Legs proportionaly. In the Triangle ABC let the Line DE, be parallel to BC: I fay that AD, is to AD as AB to A and AB: BC:: AD: DE. Alfo DE: BC: AD: AB: or AD: AB:: AE: AC. For As ABC, and DE are fimilar because <D = < Band ≤ E = <G (by the 6th) and A is common to both. their Sides about their equal Angles are proportional (by the laft) Q. E. D. Of Right Lines applied to a Circle. DEFINITIONS, 1. Every Circle is fuppofed to be divid ed into 360 D g. and each Deg. into 60 parts, called Minutes, and each Minute into 60 parts, called Seconds, &c. Any Portion of the Circumference whereof is an Arch, and is Measured by the Number of Degrees it contains. 2. Fig. 55. A Chord is a Right Line joyning the Extremities of an Arch, as AC, is the chord of the Arches ABC, ADC. 3 A Tangent of an Arch is a Righline drawn Perpendicular to the end of the Radius or Semi D'ameter, paffing through one end of the Arch, and its length is limited by a Right-line drawn for the Center through the other End of the Arch, which is called the Secant; thus BM is the Tangent, and FM the Secant of the Arches AB and AD. 4. A Right Sine is a Right-line drawn from one End of an Arch, Perpendicular to that Diameter paffing through the other End, or is half the Chord of the double Arch; AE is the Right Sine of the Arches AB, and AD. And here is evident, that the Sine of 90 Deg. which is equal to the Radius, or Semi-diameter of that Circle, is the greatest of all Sines, the Sine of au Arch greater then a Quadrant, being lefsthan the Radius. 5. A Verfed Sine is the Segment of the Diameter intercepted between the Arch, and the Right Sine, EBis the Verfed Sine of the Arch AB, and ED of the Arch AD. 6. The difference of an Arch from a Quadrant, whether it be greater or lefs, C 4 is is call'd its Complement, GA is the Complement of the Arches AB, AD; HA is the Sine of the Complement, or Cofine, GI the Tangent of that Complement, or Cc-Tangent, FI the S cant of that Complement, or Co-fecant. Plane Trigonometry, Is the Menfuration of the Sides and Argles of plain Triangles. A plain Triangle has fix parts, viz. Three Sides and three Angles, whereof any three being given, except the three Angles, the other may be found by Trigonometrical Calculation. In right Angled Triangles, there are feven Cafes, all performed by the following Axioms. AXIOM I. In any Right Angle Triangle, if either of the Legs be fuppofed to be the Radius of a Circle, the other Leg will be the Tangent of the oppofite Angle or of the Angle at the Center, and the Hypothenufe will be the Secant of that Angle: But if yon imagine the Hypothenufe to be the Radius of a Circle, then each Leg will be the Sine of its oppofite Angle, or of the Angle at the Center, as is plain from. Fig. 56.57:58:59. |