If HG and FI h: drawn through any point E of the Diigonal CB parallel to AC and CD, the Parallelogram ACDB is divided into four Parallelograms, two of which are about the Diagonal, and the other two are their Coinplements, which are thus Thewn to be qual, 4 ABC = to a CDB

( and the as HB E and EFC are = to as IBE and ECG (by 8) If from the Equal AS ABC and CDB vou subftra& the Equel AS HBE, EFC and IBE, CEG there will remain the Parall-logram AHEF = to the Parallelogram EGID (by 2d Axiom.)

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Fig.52. In every rectangle Triangle A B C, the Square of the side AC, which is opposite to the right Angle, is equal to the Squares of the other two sides (A B, CB)

Demon. D:aw IC.BF,and BE,Parallel to AF. I then you add the conmon < BAC toine right Angles IAB, FAC, and therefore e. gial, the wholes TAC, FAB, will be qial, but the 4s IAC, FAB have the two lides which contains those Angles equal (by Def.15) to wi! IA = BA and CA = FA.: IIAC

A FAB (by Prop. 1) hut A TAC = 를 thegrare ILBA and ABF= Parallelo

gram AFZ the (by Prop.10) therefore Square LIBA =the Parallelograin AFZE. I might te shewn with the fame Ease that the Square BXCH =the Parallelograin CZER. Q. E D.


Fig.53. An (BCA) at the Center is double to the < (AFB) at the Circuinference when the same Arc (AB) is Bife to both Angles. This Prop. bath three Cafts, The first is when the fide (CA) coincides with the side (AF). For then CF = CB, because both ara . drawn from the Center to the Cicchinference of the same Circle therefore in a CFB<CBE = < CFB (per Prop. 2) but <BCA => CBF t < CFB (per Schol

. 7 Prop.) ; ACB is double the <CFB which may the first Lo che second Case CA and CB tall without AF and BF. Then << C A is double < A FX, and < XCB is doubl, the <XFB (by the fifi Cafe) Thesetore the wirole < ACB is donble the whole < AEB. In the third Case RK cuts CA, and the < AKB is wholly without the < ACB Draw KCL then < ACL is double < AKL (by the filt Cafe) and if < LCB and its double <LKB be taken away there remuins <A B doub.c. FAKB. Q.E.D.

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All fimilar Triangles bave their fides about their equal Angles proportional. For if they were inscribed in Circles, their Şides would be Chords of funilar Arcs.


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 pa. sallel to BC: I say that AD, is in AD as AB to A and AB: BC:: AD: DE Allo DE: BC: AD: AB:or AD: AB :: AE: AC. For As ABC, and · DE are similar because

<D=< B and <E= <G (by the 6th) and c A is common to both .. their Sides about their qual Angles are proportional (by the luft) Q. E. D.

Of Right Lines applied to a Circle.


1. Every Circle is supposed 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. on of the Circunference whereof is an Arch,

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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 Righi. line drawn Perpendicular to the end of the Radius or Semi-Dameter, palling through one end of the Arch, and its length is li. mited by a Right-line drawn for the renter through the other End of the Arch, which is called the Secant; thus BM is the Tan. gent, 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, Perpen. dicular to that Diameter passing through the other End, or is half the (hord of the double Arch; AE is the Right Sine of the Arches AB, and AD. And here is evident, That the Sine of 90 D:g. 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 less than the Radius.

5. A Versed Sine is the Segment of the Diameter intercepted between the Arch, and the Right Sine, EB!is the Versed Sine of the Arch AB, and ED of the Arch AD.

6. The differercu of an Arch from a. Quadrant,, whether it be greater or less, 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, Fl the Scart of that Complement, or Cc-secant.

Plane Trigonometry,

Is the Menfuration of the sides and Are gles of plain Triangles. A plain Triangle has fix parts, viz. Three Sides and three Angles, whereof any three being given, ex c?pt the three Angles, the other may be tound by Trigonometrical Calculation.

in right Angled Triangles, there are Seven Cases, all performed by the following -Axioms,


In any Right Angle Triangle, if either of the Legs be supposed to be the Radius of a Circle, the other Leg will be the Tan. gent of the opposite Angle or of the Angle at the Center, and the Hypothenuse will be the Secant of that Angle : But if yon im

in agine the Hypothenuse to be the Radius of a Circle, then each Leg will be the Sine of its opposite Angle, or of the Angle at the Center, as is plain from. Fig. 56.57:58:59.

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