Page:Cyclopaedia, Chambers - Volume 2.djvu/899

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If ancient Demefne be pleaded of a Manor, and deny'd, this fhall be trfd by the Record Of Doomfday. See De- mesne and Doomsday.

Battardy, Excommunication, Lawfulnefs of Marriage, and other Ecclefiaftical Matters, fhall be trfd by the Hi- lltop's Certificate.

Before Trial in a Criminal Cafe, 'tis ufual to ask the Cri- minal how he will be try A ? which was anciently a very pertinent Queftion, tho' not fo now ; in regard there were formerly leveral Ways of Tridl, vim. by battel, Ordeals, and fury. See Battel, Ordeal, and Jury.

When the Criminal anfwer'd by God, and his Country, it fhew'd he made Choice to be trfd by a Jury. — But there is now no other Way of Trial. This is alio call'd trying per pais, per patriam

For the ancient Marnier 0/ Trial by Combat and great j4(Jize ; SeeCoMBAT, Duel, and Assize.

TRIANGLE, in Geometry, a Figure comprehended un- der three Lines, and which of Conlequencahas ihree Angles. See Figure and Angle.

If the three Lines or Sides of the triangle be all right, it is faid to be a 'Plane or Retlilinear Triangle. See Plane and Rectilinear.

If all the three Sides of rhe Triangle be equal, (as A BC Tab. Geometry, Fig. 68.) it is laid to be equilateral. See Equilateral.

If onlv two ol the Sides of the Triangle be equal, (as in DEF, Fig. 69.) it is call'd an Ifofceles, or Bglticrural Trian- gle. See Isosceles, Sjfr.

If all the Sides of the Triangle be unequal to each other, (as in ACB, Fig. 70.) the Triangle is faid to be Scalenous. See Scalenous.

If one of the Angles (as K of Fig. 71.) of a Triangle KML be a right Angle, the Triangle is laid to be Retlan- gular. See Rectangle.

If one of the Angles (as N Fig. 72.) be obtufe, the Tri- angle is faid to be Obtifangular, or Amblygonous. See Ob-

TUSANGLE.

If all the Angles be acute, (as in ACB, Fig.68.) the Triangle is laid to be pentangular, or Oxygenous. See

AcUTANGLE, &C.

If the three Lines of the Triangle be all curves, the Tri- angle is faid to be Curvilinear. See Curvilinear.

If fome of the Sides be right, and others curve, the Angle is ftid to be Mixtilinear. See Mixtilinear.

If the Sides be all Arches of great Circles of the Sphere, the Triangle is laid to be Spherical. See Spherical Tri- angle.

Cmfirttatm of Triangles.

1°. Two Sides, as A Band AC, Fig. 73. being given in Numbers,orotherwife, together with the Quantity of theAngle intercepted between them, A ; to conllruci a Triangle — Affiime A 15 as a Bale ; and in A make the given Angle : On the other Leg let off the other given Line AC 5 lallly, draw B C : Then will A B C be the Triangle requir'd.

Hence, two Sides with the intercepted Angle being de- termin'd, the whole Triangle is detetmin'd. — Wherefore, if in two Triangles ACB and acb; a=^A ; and ab : ac : : A B : AnC, the Triangles are determin'd in the iame Manner, and are therefore fimilar ; confequently c = C and b=B, ab -.be. : AB: BC, r$c.

1°. Three Sides, AB, BC, andC A, Fig. 70. being given, any two whereof, as AC, AB, taken together, are greater than the third, to conftrucf. a Triangle — AfTume A B for a Baft ; and from A, with the Interval AC, defcribe an Arch y j and from B, with the Interval BC, defcribe another Arch x 1 Draw the tight Lines AC and BC. Thus is the Triangle conftructed.

Hence, as of any three given right Lines, only one Tri- angle can be constructed ; by determining the three Sides, the whole Triangle is determin'd.

Wherefore if in two Triangles ACB and acb ; AC: AB: ■■ ac : ab $ AC: CB :: ac : be ; the Triangles are de- tetmin'd in the lame Manner, and confequently are fimilar, and therefore are mutually equiangular.

o°. A right Line, as AB, and two adjacent Angles A and B which, taken together, are lefs than two right ones, being given; to defcribe the Triangle ABC. — On the given Line AB, make the two given Angles A and B : Continue theSides'AC and BC till they meet in C. Then will ABC be the Triangle requir'd.

Hence, one Side and two Angles being given, the whole Triangle 'is determin'd. — Wherefore if in two Triangles A=a and B = i, Fig. 73. the Triangles are determin'd after the fame Manner, and therefore are fimilar.

Menfuration cf Triangles.

To find the Area of a Triangle. - Multiply the Bafe A B Fig. 74. by the Altitude Cd ; half the Product is the Area of

the Triangle ABC.

Or rhus : Multiply half the Bafe A B by the Altitude CD ; or the whole Bafe by half the Altitude ; the Product, is the Area of the Triangle.

IS.gr. AB=34i

Ci = 2J +

I3«8 I026 684

2) 80028 (

AB = 3 4 2 I CD = ii 7

=>394 34* 34*


 * ab=i 7 i

0,^ = 234

684 5'3

34*

40014

40014 Area

are as

40014

Or, The Area of any Triangle is had by adding all the three Sides together, and taking half the Sum 3 and from that half Sum, fubitracting each' Side feverally, and multi- plying that half Sum and the Remainder continually into one another, and extracting the Square Root of the Product.

Hence, 1°. If between the Bale, and half the Altitude'; or between the Altitude, and half the Bafe, be found a mean Proporrional ; it will be the Side of a Square equal to the Triangle.

2°. If the Area of a Triangle be divided by half the Bafe, the Quotient is the Altitude.

General 'Properties of 'Plane Triangles.

i°. If in two Triangles ABC and abc. Fig. 73. the Angle A£c = <i; and the Sides AB = ab, and AC=« ; then will theSidesBC = *<;, and C = c, theAngleB=£; and there- fore the whole Triangles will be equal aiid fimilar.

2 . If one Side of a Triangle ABC (Fig. 75.) be conti- nu'd to D; the external Angle DAB will be greater than either of the, internal oppofite ones B or C.

3 . In every Triangle, the greatelt Side is oppos'd to the greateft Angle, and the leaft to rhe leaft,

4°. In every Triangle, any two Sides taken together are greater than the third.

5°. If in two Triangles, the feveral Sides of the one be reflectively equal to the Sides of the other, the Angles will likewife be reflectively equal ; and confequently the whole Triangles equal and fimilar.

6°. If any Side, as A C (Fig. 76". J of a Triangle A CB be continu'd to D, the external Angle DCA will be equal to the two internal oppofite ones^ and z taken together.

7 . In every Zriangle, as ABC, the three Angles A, B, C, taken together, are equal to two right ones, or i8o Q.

Hence, 1°. If the Triangle be rectangular, as M K L, (Fig. 7 1.) the two oblique Angles M and L, taken together, make a right Angle, or 90 ; and therefore are half Right, if the Triangle be Ifofieles. — 2 . If one Angle of a Tri- angle be oblique, the other two taken together are oblique likewife. — 3 . In an Equilateral Triangle, each Anole is 6o° 4. If one Angle of a Triangle be fubllraaed°from l8o c. the Remainder is the Sum of the other two ; and if the Sum of two be fubllracted from 180, the Remainder is the third. — 5°. If two Angles of one Triangle be equal to two of another, cither togerher or feparately ; the third of the one is likewife equal to the third of the other. — 6° Since in an Ifofceles Triangle DFE (Fig. 69.) the Angles at the Bafe^ji and v are equal ; if the Angle at the Vertex be fub- ftrafled from 180°, and the Remainder be divided by 2, the Quotient is the Quantity of each of the equal Angles : In like Manner, if the double of one of the Angles at the Bale y be fubllracted from 180, rhe Remainder is "the Quantity of the Angle at the Vertex.

8". If in two Triangles, ABC, and abc, Fig. 73. AB =

ab, A=aand B=i ; then will AC=«,B(i=i[, C=tr and the Triangle ACB equal and fimilar to the Triangle abc — Hence, if in two Triangles, ACB and acb, A=a, B = £and BC=£c; then will C=c ; confequently A C=

ac, AB =ab ; and the Triangle ACR=acb.

9°. If in a Triangle D FE the Angles at the Bafe y and v, Fig. 69. be equal, the Triangle is Ifofceles : Confequently, if the three Angles be equal, it is equilateral.

io°. If in a Triangle ABC, (Fig. 77.) a right LineDE be drawn parallel to the Bafe; then will B A : BC ■ • BD ■ BE:. -AD: EC. AndBA: AC::BD:DE. Confe- quently the Triangle B D E fimilar to B A C.

1 1°. Every Triangle is infcribable in a Circle. See Circle.

12°. The Side of an equilateral Triangle, inferib'd in a Circle, is in Power triple of the Radius. See Raltus.

1 3 . Triangles on the fame Bafe, and having the fame Height, that is between the fame parallel Lines,' are equal See Parallel. Y ■ - "

14 . Every Triangle, asCFD, Fig.41. is one half ofa Pa- rallelogram ACDB on the fame, or an equal Bale CD and of the fame Altitude, or between the fame Parallels : Or a Triangle is equal to a Parallelogram upon tho fame Bale, but half the Altitude ; or half the Bafe, and the fame Altitude. See Parallelogram.

15°. In