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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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