T R I
[ ¥1
t R I
From one of the given Angles B, letting fall the Perpendi- cular B E, to the oppofite Side A C ; in the reftangled Triangle ABE, from the given Angle A, and Hypothenufe A B, we find the Angle ABE; which fubtrafled from A B G, leaves the Angle EBC. In cafe the Perpendicular fall without the Triangle, ABC is to be fubtrafled from ABE; fines by afluming B E for a lateral Part in the Triangle C E B, the Angle G is the middle Part, and the Angle CBE the dis- joint Part ; and in the Triangle ABE, the Angle A is the middle Part, and the Angle ABE the disjoynt Part : The Gofine of the Angle C is found by fnbtraffing the Sine of the Angle ABE, from the Sum of the Coiine of the Angle A, and of the Cofine of E B C-
10°. Given tin Angles A 43° 20', and C 82° 34', toge- ther with a Side B A 66° 45', oppofite to me of them ; to find the other Angle.
From the fought Angle B, let fall a Perpendicular B E ; and in the right angled Triangle A E B, from the given Angle A, and Hypothenufe BA, find the Angle ABE; fince af- firming the Perpendicular E B for a lateral Part in the Trian- gle E C B, the Angle E is the middle Part, and the Angle CB E a disjunct Parr ; and in the Triangle A B E, the Angle A is the middle Part, and the Angle A B E a disjunft Part : The Sine of the Angle E B C is found by fubtraftlng the Cofine of A from the Sum of the Cofine of C, and of the Sine of A B E. If then A B E and E B C be added, or in cafe the Perpendicular fall without the Triangle, be fubtract- ed from each other, the Refult will be Angle requir'd A BC.
n°. Given the three Sides ; to find an Angle oppofite to one ~cf them.
I. If one Side A C, Fig. 16. be a Quadrant, and the Leg A B lefs than a Quadrant, find the Angle A. Continue AB to F, till AF become equal to a Quadrant \ and from the Pole A draw the Arch C F, to cut the Arch A F at right Angles in F. Since in the reflangled Tri : vgle C R F, we have given the Hypothenufe B C, and the Side A F, or its Complement A B to a Quadrant ; we Shall find the Perpendicular C F C, which being the Meafure of the Angle CAB, that Angle is found of courfe.
II. If one Side A C be a Quadrant, and the other AB greater than a Quadrant, feek again the Angle A : From a B fubtracr. the Quadrant A D, and from the Pole A defcribe the Arch CD, cutting the Arch CD at right Angles in D. Since in the reftangled Triangle CDB, the Hypothenufe B ', and Side DB, or Excefsof the Side A B beyond a Quadrant, is given, the Perpendicular CD will be found as before, which Is the Meafure ot the Angle A requir'd.
HI. If the Triangle be Ifofceles, andAC = CF, and the Angle A C F be requir'd ; biffeft A F in D, and thro' D and C draw the Arch D C. Since CD is perpendicular to A F, the Angles A and F, and A C D and D C F are equal ; from the Hypothenufe A C, and Leg A D, given in the recfangled Triangle A C D, we find the Angle ACD ; the double whereof is tie Angle requir'd ACF : and from the fame Data may the Angle A or F be found.
I'V. If the Triangle be Scalenous, and the Angle A, Fig. 32. be requir'd ; from C let fall the Perpendicular CD, and feck the Semi-difference of. the Segments AD and D B, by faying, As the Tangent of half the Bafe A B, is to the Tan- gent of half the Sum of the Legs AC and C B ; fo is the Tangent of their Semi-difference, to the Tangent of the Semi-difference of the Segments A D and D B : Add then the Semi-difference of the Segments to the half Bafe, to find the greater Segment ; and f'ubtrafl the fame from the fame for the lefs. Thus having in the reftangled Triangle CAD, the Hypothenufe AC, and the Side A D, the Angle A is found thence. After the fame Manner is B found in the other CDB, from C D and D B given.
ii°. Given the three Angles A B and C, to find any of the Sides.
Since in lieu of the given Triangle, another may be af- fum'd, whofe Sides are equal to the given Angles, and the Angles to the given Sides ; this Problem is refolv'd after the lame Manner as the preceding one.
Triangle, Triangulum, in Afrronomy, a Name common to two Confkllations ; the one in the Northern Hemifphere, call'd fimply Triangulum, or Triangulum Cm- lejie ; the other in the Southern Hemifphere, call'd Triangu- lum AuflraJ.e. See Constellation.
The Stars in the Northern Triangle, in •Ptohmy's Cata- logue ate 4 ; in Tycho's as many ; in the Britannic 24 : The Longitudes, Latitudes, Magnitudes, &C. whereof, are as follow :
Stars in the Cancellation Triangulum.
Nuiffii i"id Situation of tin Stars.
That preceding the Vertex Vertex of rhe Triangle That following the X'ertex lirtlof 3 intheBsfe
^Longitude
o of 17 2 30 p
6 00 2f
7 f9 44 7 13 4;
^atituc
e*.
'
>t
17 39
08
rt> 48
n
19 28
00
20 34
17
17 06
18
Na?xet and Situation of ^ the Star:. ^
ift of 3 Inform, under Triangle Contig. to the laft of the Bale
Middle one of the Bale
Laft of the Bale
S. oif Inform, under theTriang.
Laft of thefe Informes
A fmalier contiguous to If,
if
Informes between the Triangle and the Rams Tail,/ which arc alio minibcr'd among the Stars of Aries.
^ongituue
011*
6 r2 3f
8 42 40
9 09 43 9 10 21
7 53 3"
jo 32 f2 9 S9 if 10 12 15- 13 08 28
13 ir 01
10 14 if
11 48 01 <* 3j- 47 J? f 1 4f 16 15 ,-3
It) 22 2f
16 39 24
18 37 f6 18 41 07
Latitu.ee.
1 a
If f 9 C2
18 34 12
19 21 32
18 f6 07
15 ss 16
16 16 32
14 13 08
14 24 24
20 00 37
18 26 iS
8 49 4S
10 fl f2
11 17 13
10 2f 37
8 f ■ U
8 59 42
7 29 04
10 f4 26
8 f8 28
rzlth s 3>ftf
40tll
48th Wa- j~
( ries
yoth
fift
f4th
Ijfth)
TRIANGULAR CcmfflJJh, are fuch as hare three Legs or Feet, whereby to take off any 'Triangle at once. See Compasses.
Theie are much us'd in the Conftruction of Maps, Globes,
Triangular Numbers, are a kind of Polygonous Num- bers ; fee Polygonou.'. Number $ being the Sums of Arith- metical Progreffions, the Difference of whofe Terms is i z Thus,
Of Arithmetical Progrefs 123456 are form' d Tnavg. Numb. [ 3 6 10 15 21.
Triangular £Hiadrant t is a Sector furnifh'd with a loofe Piece, whereby to make it an Equilateral Triangle* See Sector.
The Calendar is graduated thereon, with the Sun's Place, Declination, and other ufeful Lines j and by the Help of a String and a Plummet, and the Di villous graduated on the looie Piece, it may be made to icrve for a Quadrant. See Quadrant.
TRIANGULARIS, in Anatomy, a Name given to two Mufcles, in refpecf of their Figure. See Muscle.
The Triangularis iPe£fc?is, which has fomerimes the Ap- pearance of three or four difiincl Mulcles, ariles from the lnfide of the Sternum, and is implanted into the Cartilages which join the four loweft true Ribs ro the Sternum.
The Action of this Mufcle is very obfeure ; fince both the Origination and Infertton are at Parts not moveable, bu-c together. — Dr. ^Drafts conjectures it may conduce towards the forming the neceffary Incurvation ot the Sternum, and by its Over-tenfion in Children, while the Cartilages are loft, may occafion that morbid Acumination of the Sternum (eeh in rickety Children. — Others fuppofe it may contrail: the Cavity of the Thorax in Expiration.
Triangularis L&biu See Depressor Labii Su$e~ rioris.
TRIARII, in the Roman Militia, a kind of Infantry, ann'd with a Pike and a Shield, a Helmet and a Cuirafs.
They were thus call'd, becaule they made the third Line of Battle.
'Polybius diftingui fries four Kinds of Forces in the Roman Army : The firft, call'd ^Pildti, or Velites, were a raw Sol- diery, lightly arm'd. — The Hafiati, or Spear-men, were a Degree older, and more experiene'd. — The third, call'd IPrincifeSj Princes, were ft ill older, and better Soldiers than' the fecond. — The fourth were the eldelf, the moft experi- ene'd, and the braveft : Thefe were always difpos'd in the third Line, as a Corps de Referve, to fiilUin the other two, and to reftore the Battle, when the others were broken or defeated.
. Hence their Name of Tfi&fti ; and hence the Proverb tt£ Triarios ventum eji, to mew that one is at the laft and hard- er! Struggle.
The Triarii were alfo call'd Toftpgnanu becaufe rang'd behind the (PfinciJ>es, who bore the Standard in a Legion, See Principes.
TRIAS Harmonica, or the flatmonicd T»rAi>i in Mafic, a Compound of three radical Sounds, heard all together 5 two whereof are a Fifth, and a Third above the other, which is the Fundamental. See Concord, ££c.
The Triad is properly a Conibnance form'd of a Third and a Fifth 5 which, with the Baft, or fundamental Sound, makes rhree different Terms, whence the Name c Jrias. — That of harmonical is doubtlefs given it from that wonderful Proper- ty of the Fifth, which divides itfelf naturally into two Thirds, both excellent, and perfectly harmonical; fo that this one
Sound
