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Chambers, Ephraim, 1680 (ca.)-1740 / Cyclopædia, or, An universal dictionary of arts and sciences : containing the definitions of the terms, and accounts of the things signify'd thereby, in the several arts, both liberal and mechanical, and the several sciences, human and divine : the figures, kinds, properties, productions, preparations, and uses, of things natural and artificial : the rise, progress, and state of things ecclesiastical, civil, military, and commercial : with the several systems, sects, opinions, &c : among philosophers, divines, mathematicians, physicians, antiquaries, criticks, &c : the whole intended as a course of antient and modern learning
(1728)

Tribunus - tything,   pp. 248-270 PDF (22.0 MB)


Page 249


Ir IL1                             2
They conlii of two entire Gutters or Channels, cut to a
right Angle, call'd Glyphes, and feparated by two Initrfices,
call'd, by lirruvius, Shanks, from two Half Channels at
the Sides. See GLYP HES.
The ordinary Proportion of Triglyphs, is to be a Module
broad, and one and a Half high. ---- .1ut this Proportion,
M. le Clerc obferves, fometimes occafions ill-proportion'd
Intercolumnations in Portico's; for which Reafon he chufes to
accommodate the Proportion of his Triglsypbs to that of the
Intercolumns. See INTERCOLUMNATION.
The Intervals between the Triglyphs, are called Metopes.
See MEYoPEs.
Under the Channels, or Glyphes, are placed Gutte or Drops.
See GUTTJE.
The 7riglyphs make the moil diflinguifhing Charadfer of
the Doric Order. --- Some imagine them originally intended
to convey the Guttm that are underneath them: Others fancy
they bear fome Refemblance to a Lyre, and thence conjeaure
the Order to have been originally invented for fome Temple
facred to Apollo. See DoRic.
The Word is form'd from the Greek 7p1)Av90t, three En-
gravings. from -Abyss, scultO.
TOrGON, TRIGONUS, in Geometry, aTriangle. See
TRIANGLE.
The Word is form'd from the Greek 7'pV(ol-, Triangle.
TRIGON, in Aftrology, is an Afpecd of two Planets,
wherein they are 120 Degrees diflant from each others
called alfo Trine. See TRINE.
The Brigons of Mars and Saturn, are held Malefic Afpects.
See AsPEcT.
TRIGON, TRIGONON, in Mufic, is a Mufical Infirument,
ufed among the Ancients.
The 5Jriion was a kind of triangular Lyre, invented by
Ibycus. See LYRE.
TRIGONOMETRY, 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 ref+ 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 witch the
third Side B C, Tab. Trigonometry Fig. r.
The Word literally fignifies the meafuring of friangles;
form'd from the Greek, rayPv , Triangle, and etfov, Mea-
fure; yet does not the Art extend to the meafuring of the
Area or Sarface of Triangles, which comes under Geometry:
?'rigonometry only confiders the Lines and Angles thereof.
Tlrigonometry is of the utmoff Ufe in various mathematical
Arts. -- 'Tis by means hereof, that moff of the Operations of
Geometry and Affronomty are perfiorm'd; without it the Mag.-
nitude of the Earth and the Stars, their Diflances, Motions, E-
cipfes, Fec. would be utterly unknown.-- Trigonometry, there-
fnre, muff be own'd an Art, whereby the moft hidden Things,
and thofe remotefi from the Knowledge of Men, are brougit
to light. A Perfon ignorant hereof, can make no great Pro-
grefs in mixt Mathettaticks; but will often be gravell'd, even
in Natural Philofophy, particularly in accounting for the
Phxnomena of the Rain-bow, and other Meteors.
7Trigonometry, 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, Rangents and Secants.
See RADITS, CHORD, SINE, TANGENT and SECANT.
This Proportion of the Sines and Tangents to their Radius,
is fometimes exprefs'd in common or natural Numbers, which
confsitute what we call the Tables of naturalSines, Tangents,
Ac. --. Sometimes it is exprefs'd in Logarithms, and in that
Cafe, conflitutes the Tables of artifetal Sines, Oc.  See
TA BLES.
Lafily, fometimes the Proportion is not exprefs'd in Xum-
bers; but the feveral Sines, Tangents, Ac. are aa5ually laid
down upon Lines or Scales; whence the Line of Sines,
Tangents, Uec. See LINE and SCALE.
Trigonometry is divided into Plain and Spherical: The
firfi: confidering reailinear Triangles; and thefecond, Spheri-
cal-ones. --- The Firfl is of obvious and continual Ufe in Navi-
gation, Meafuring, Surveying and other Operations of Geo-
merry. See MEASJURING, SURVEYING, SAILING, FC.
The Second is only learn'd, with a View to Affronomy and
its kindred Arts, Geography and Dialling. -- It is generally
efleem'd exceedingly difficult,by reafon of the vafi Number of
Cafes wherewith it is perplex'd; but the excellent Wolfius has
removed moft of the Difficulties. That Author has not only
fhewn how all the Cafes of reolangled Triangles may be
folv'd the common Way, by the Rule of Sines and Tangents ;
but has likewife laid down an univerfal Rule, whereby all
Problems, both in Plain and Spherical redtangled Triangles
are folved: And even obliquangular Triangles, he teaches to
folve with equal Eafe. --- His Doarine, fee under the Article
TRIANGLE.
'Plain TRIGONOMETRY, is an Art whereby, from three
given Parts of a Plain Triangle, we find the reft.
49]I
TR I
The great Principle of Plain lfrigovometry, is, that iTT
every plain Triangle, the Sides are, as the Sines of tHe oppofite
Angles. ---- See this Principle applied to the Solution of the
feveral Cafes of Plain Triangles, under the Article TR I-
ANGLE,.
Spherical TRIGONOMETRY, -s the Art whereby, from
three given Parts of a fpherical Triangle, we find the refl;
E. gr. whereby from  two Sides and one Angle, we find the
two other Angles, and the third Side.  See S IaPR I CI S.
The Principles of Spherica    ii oizerry, as reform'd by
Woljius, are as follow:
IO In every reaangledfpterical 7riaigge, AZ C, relaregzar
at C, the whole Sine is to the Sine of the Itpotbenzufe, B C,
Tab. rrigonom. Fig. 3 ,. as the Sine of either of the acute
Angles, as C, is to the Sine ef the Ltg, oplofite thereto A B ;
Or, the Sine of the Angle .B, to the Sine of its olpcfite Leg A C5
whence we deduce, that the Red1angle of the whole Sine into
the Sine of one Leg, is equal to the Redangle of the Sine of
the Angle oppofite thereto, into the Sine of the Hypothenufe.
2f In every right- angled fherical Triangle A  i C, Fig. 3 1.
ozone of whofe Sides is a Quadrant ; if the Complements of the
Legs AfB and AC to a k-jtadrant, Ie confider'd as the Legs
themfelves; the Reflangle of the whole Sine into the Coftne of
the middle 'Part, is equal to the RefLangle of the Sines of the
disjunt7 orfeparate Parts.
Hence, l . 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 disjund Parts.
0--- 2 Since, in the re'illinear Triangle A B C (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 A C or A B: if,
inflead of the Sines of the Sides, we take the Sides themfelves;
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 disjund Parts
B or C and B C; i. e. to the Sine of B or C, and B C itfelf.
This, TI'olfus calls Regula Sinium Catholica, or the firfi
Part of the Catholic Rule of Trigonometry; by means where-
of all the Problems of either Trigonometry are folved, when
the thing is effie~ed 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
B and 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 disjun&, or, as he calls them, oppofite
Parts. --- But, in this, that Harmony between Plain and Sphe-
rical Sirigcnometry, vifible in Wqioluus's Rule, does not appears
30 In a relangled fpherical Triangle AS? C (Fig. 3;.) none
of wvhofe Sides is a Quadrant; as the whole Sine is to the Sine
of the adjacent Leg A C; fo is the Tangent of the adjacent
Angle C to the langent of the Leg Al.
Whence, i f, 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 A B; 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. 20 The Re~angle, therefore, ofthe whole
Sine, into the Sine of one Leg A C, is equal to the Redangle
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 Redangle of the whole Sine, into the Sine of the Leg
A B,. is equal to the Redangle of the Tangent of the Leg
A C into the Co-tangent of the Angle B.
40  n everyright-angledfpherical ThangleAiBC (Fig. 31)
vone of whofeSidcs is a ,zuadrant; if the Comflements of the
Legs ABd an.d   C to a Oyadrant, or their Exceffes t-eyond a
Quadrant, becon derd as the Legs tPenifelves i the hefangle
o: the whole Sine, into the Co-fine of the middle Part, will
be equal to the Relaagle cf the Gc-tanegents, of the conjtntl
Parts,
Hence, If, 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 thecontiguous Parts. 20 Since in a redi-
linear, right-angled Triangle, we ufe the Tangents, when
from the Legs AB and AC (Fig. 34.) given, the Angle C
is to be found; and in that Cafe the whole Sine is to the
Co-tangent of C. i. e. to the Tangent of B, as A B to A C
therefore, alfo, in a rectilinear TrIangle, if for the Sines and
Tangents of the Sides be taken the Sires themfelves; the
whole Sine, with the Co-fine of the middle Part, i. e. with
A C, is equal to the Co-tangents of the conjundt Parts, i. e.
to the Co-ta          C, or Tangent of B and the Side
This, Wofius calls Reglyla T-a ventium Cathoiica, and con_
flitutes the other Part of the Catholic Rule of Trigovometry;
whereby all Problems in each T'rigonmcretry, where Tangents
are required, are folved.
My Lord Neper'% Rule to the like EfFed, 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 Vrig.
7ometrty; that in a reflangled Triavgle, (notatis Xotagdis, i. e.
LRrrJ


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