Page:Cyclopaedia, Chambers - Volume 2.djvu/908
T R I
C H9 ]
T R I
They confift of two entire Gutters or Channels, cut to a right Angle, call'd Glyphes, and feparated by two Interftices, call'd, by Vitrtivius, Shanks, from two Half Channels at the Sides. See Glyfhes.
The ordinary Proportion of T'riglyphs, is to be a Module broad, and one and a Half high. — But this Proportion, M. le Clerc obferves, fome times occafions ill-proportion'd Intercolumnations in Portico's 3 for which Reafon he chufes to accommodate the Proportion of his Triglypbs to that of the Intercolumns. See In t er colu m n at ion .
The Intervals between the Trtglyphs, are called Metopes. See Metopes.
Under the Channels, or Glyphes, are placed Guttse or Drops. See Guttje.
The Trig/yphs make the moft diftinguiming Character of the T)oric Order. — Some imagine them originally intended to convey the GutfE that are underneath them : Others fancy they bear fome Refemblance to a Lyre, and thence conjecture the Order to have been originally invented for fome Temple facred to Apollo. See Doric.
The Word is form'd from the Greek TfiyKvipQ-, three En- gravings, from yhvyu, Sciilpo.
TRIGON, Trigonus, in Geometry, a Triangle. See Triangle.
The Word is form'd from the Greek Tpiyw©-, Triangle.
Trigon, in Aftrology, is an Afpect of two Planets, wherein they are 120 Degrees diftant from each other 5 called alfo Trine. See Trine.
The Trigons of Mars and Saturn, are held Malefic Afpects. See Aspect.
Trigon, Trigonon, in Mtijic, is a Mufical Inftrument, uled among the Ancients.
The Trigon was a kind of triangular Lyre, invented by Ibyciis. See Lyre.
TRI GONOMETRY, the Art of finding the Dimenfions of the Parts of a Triangle unknown, from other Parts known : Or, the Art whereby, from any three Parts of a Triangle given, all the reft are found. See Triangle.
Thus, e.gr. from two Sides A B and A C and an Angle B, we find by Trigonometry, the other Angles B and C with the third Side B C, Tab. Trigonometry Fig. 1.
The Word literally fignifies the measuring of Triangles; form'd from the Greek, rflyav©-, Triangle, and jmj^qv, Mea- fure ; yet does not, the Art extend to the meafuririg of the Area or Surface of Triangles, which comes under Geometry : "Trigonometry only confiders the Lines and Angles thereof.
Trigonometry is of the utmoft Ufe in various mathematical Arts. — 'Tisby means hereof, that moll of the Operations of Geometry and Aftronorhy are per form'd 3 without it the Mag- nitude of the Earth and the Stars, their Diftances, Motions, E- clipfes, Effc. would be utterly unknown.-- Trigonometry, there- fore, muft be own'd an Art, whereby the moil hidden Things, and thofe remoteft from the Knowledge of Men, are brought to iiaht. A Perfbn ignorant hereof, can make no great Pro- gress in mixt Mathernaticks ; but will often be gravell'd, even in Natural Philofophy, particularly in accounting for the Phenomena of the Rain-bow, and other Meteors.
Trigonometry, or the Solution of Triangles, is founded on that mutual Proportion which is between the Sides and Angles of a Triangle; which Proportion is known, by find- ing the Proportion which the Radius of a Circle has to cer- tain other Lines, coll'd Chords, Sines, Tangents and Secants. See Radius, Chord, Sine, Tangent and Secant.
This Proportion of the Sines and Tangents to their Radius, is fometimes expreis'd in common or natural Numbers, which coniiitute what we call the Tables of natural Sines, Tangents,
l§c Sometimes it is exprefs'd in Logarithms, and in that
Cafe, conftitutes the Tables of artificial Sines, $$o. See Tables.
Laftly, fometimes the Proportion is not exprefs'd in Num- bers 5 but the feveral Sines, Tangents, &c. are actually laid down upon Lines or Scales 3 whence the Line of Sines, Tangents, Qfc. See Line and Scale.
Trigonometry is divided into Tlain and Spherical : The firft considering rectilinear Triangles ; and thefecond, Spheri- cal ones. — The Firfi is of obvious and continual Ufe in Navi- oation, Meafuring, Surveying and other Operations of Geo- metry. See Measuring, Surveying, Sailing, &c.
The Second is only learn'd, with a View to Aftronomy and its kindred Arts, Geography and Dialling. — It is generally efteem'd exceedingly difficult, by reafon of the vaft Number of Cafes wherewith it is perplex'd ; but the excellent Wolfius has removed moil of the Difficulties. That Author has not only fticwn how all the Cafes of rectangled Triangles may be folv'd the common Way, by the Rule of Sines and Tangents 3 but has likewife laid" down an univerfal Rule, whereby all Problems, both in Plain and Spherical rectangled Triangles are folved : And even obliquangular Triangles, he teaches to folve with equal Eafe.-— His Doctrine, fee under the Article Triangle.
(Plain Trigonometry, is an Art whereby, from three given Parts of a Plain Triangle, we find the reft.
The great Principle of <Plain Trigonometry, is, that in every plain triangle, the Sides are, as the Sines of the oppofite Jingles --See this Principle applied to the Solution of the feveral Cafes of Plain Triangles, under the Article Tri- angle.
^er/^ T&igonometrv, 3s the Art whereby, from three given Parts of a fpherical Triangle, we find the reft, E. gr. whereby from two Sides and one Angle, we find the two other Angles, and the third Side. See Sphericks.
The Principles of Spherical Trigonometry, as re form'd by Wolfdis, are as follow :
1° In every retlangled fpherical Triangle, AS C, reBargvlar atC, the while Sine is to the Sine of the Hypothenufe S C, Tab. Trigmom. Fig. 33. as the Sine of either of the' acute Angles, as C, is to the Sine of the Lrg, oppofite thereto A % 5 Or, the Sine of the Angle S, to the Sins of in oppofite Leg A C ; whence we deduce, that the Rectangle of the whole Sine into the Sine of one Leg, is equal to the Rectangle of the Sine of the Angle oppofite thereto, into the Sine of "the Hypothenufe.
2 In every right- angled fpherical Triangle ABC, Fig. 31. none oftvhofe Sides is a £Hiadrant ; if the Complements of the Legs AS and AC to a Quadrant, le confiderdas the Legs tbemfelves 5 the Retlangle of the whole Sine into the Cofine of the middle 'Part, is equal to the Retlangle of the Sines of the disjuntl or feparate Tarts.
Hence, 1°. If the Sines be artificial, that is, the Logarithms of the natural ones ; the whole Sine, with the Co-fine of the middle Part, will be equal to the Sines of the disjunct Parts. — 2 Since, in the reailinear Triangle ABC (Fig. 34.) the whole Sine is to the Hypothenufe B C, as the Sine of the Angle B or C to the Sine of the oppofite Leg AC or A B : if, inftead of the Sines of the Sides, we take the Sides themfelves 5 here, too, the whole Sine, with the Co-fine of the middle Part A C or A B, will be equal to the Sines of the disjunct Parts B or C and B C 3 i. e. to the Sine of B or C, and B C itfelf.
This, WotfhiS calls Regzda Sinuum Catholica, or the firft Part of the Catholic Rule of Trigonometry 5 by means where- of all the Problems of either Trigonometry are fttlved, when the thing is effected by Sines alone. — - My Lord Neper had the firft Thought of fuch a Rule : But he ufed the Comple- ments of the Hypothenufe B C (Fig. 22.) and the Angles Band C for the Hypothenufe, and Angles themfelves: So that the Tenor of his Catholic Rule of Sines is this :
The whole Sine, with the Sine of the middle Part, is equal to the Co-fines of the disjunct, or, as he calls them, oppofite Parts. — But, in this, that Harmony between Plain and Sphe- rical Trigcnometry, vifible in Wolfius\ Rule, does not appear.
3 In a retlangled fpherical Triangle AS C (Fig. 3 3.) none ofwhofe Sides is a Quadrant 3 as the whole Sine is to the Sine of the adjacent Leg A C 5 fo is the Tangent of the adjacent Angle C to the Tangent of the Leg AS.
_ Whence, i°, as the Co-tangent of the Angle is to the whole Sine, as the whole Sine is to the Tangent of the Angle C, fo is the Sine of A C to the Tangent of ABj therefore the Co-tangent of the Angle C, will be to the whole Sine, as the Sine of the Leg adjacent thereto, A C, is to the Tangent of the oppofite one A B. i° The -Rectangle, therefore, of the whole Sine, into the Sine of one Leg A C, is equal to the Rectangle of the Tangent of the other Leg A B, into the Co-tangent of the Angle C, oppofite to the fame. And, in like manner, the Rectangle of the whole Sine, into the Sine of the Leg A B, is equal to the Rectangle of the Tangent of the Leg A C into the Co-tangent of the Angle B.
4 Q Inev?ry right-angled fpherical Triangle, AS C (Fig. 31 ) none of whofe Sides is a Quadrant 5 if the Complements of the Legs AS and AC to a§>uadrant, or their Exceffes beyond a Quadrant, beconfider'd as the Legs themfelves ; the Retlangle of the -whole Sine, into the Co-fine of the middle Tart, will be equal to the Retlangle of the Cc-tangents, if the conjunct Tarts.
Hence, i°, If the Sines and Tangents be Artificial ; the whole Sine, with the Co-fine of the middle Part, is equal to the Co-tangents of the contiguous Parts. 2 Since in a recti- linear, right-angled Triangle, we ufe the Tangents, when from the Legs A B and A C (Fig. 34.) given, the Angle C is to be found 3 and in that Cafe the whole Sine is to the Co-tangent of C. i. e. to the Tangent of B, as ABtoAC; therefore, alfo, in a rectilinear Triangle, if for the Sines and Tangents of the Sides be taken the Sines themfelves; the whole Sine, with the Co-fine of the middle Part, i. e. with AC, is equal to the Co-tangents of the conjunct Parts, /". e. to the Co-tangent of C or Tangent of B and the Side AB.
This, Wolfius calls Rrgttla Tangent'mm Catholica, and con- ftitutes the cither Part of the Catholic Rule of Trigonometry 5 whereby all Problems in each Trigontmetry, where Tangents are required, are folved.
My Lord Neper's Rule to the like Effect, is thus : — That the whole Sine, with the Sine of the middle Part, is equal to the Tangents of the contiguous Parts.
'Tis, therefore, a Catholic Rule, which holds in all Trigo- nometry ; that in a retlangled Triangle, (mtatis notandis, i. e. [ B. r r J ;he