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inhabited the country since called Dauphine. TRIESTE, a province of the Austrian empire Liv. 21. c. 31.

containing the southern half of the kingdom of TRIDAX, in botany, trailing star wort of Illyria, and burdering on the Adratic, Croatia, Vera Cruz, a genus of plants in the class of and the government of Laybach. Its territorial syngenesia, and order of polygamia superflua; extent is 5020 square miles, and its population and, according to the natural method, ranking in 540,000. The majority are Sclavonians, but the thirty-second order, papilionaceæ. It is a there are among them many Italians, Germans, native of Vera Cruz, and has flowers in the forin and Jews. The surface is for the most part of butterflies.

hilly, and the soil is, with the exception of some TRI'DENT, n. s. Fr. trident ; Lat. tridens. very fertile valleys, chalky, dry, stony, and reA three-forked sceptre of Neptune.

quires a very toilsome cultivation. The Save His nature is too noble for the world :

forms the northern boundary. The products of He would not flatter Neptune for his trident. this province are vines, olives, silk, and in

Shakspeare. general the fruits of the south of Europe. The Canst thou with fisgigs pierce him to the quick ? sheep are commonly of a good breed; but of Or in his skull thy barbed trident stick ?

corn, the quantity raised is not large. The coast Sandys on Job.

affords extensive fisheries, and the climate admits He lets them wear their saphire crowns, of making salt by evaporating the water of the And wield their little tridents.


sea. The province is divided into the four circles TRIDENTUM, an ancient town of Gallia of Trieste, Goritz, Fiume, and Carlstadt. Cisalpina, now called Trent. See TRENT. Trieste, a circle of the government of the

TRIENNIAL, adj. Fr. triennal; Lat. trien- same name, in Illyria. It contains 1440 square nis. Lasting three years.

miles, with 176,000 inhabitants, and is divided I passed the bill for triennial parliaments.

into the four arrondissements of Trieste, Duins,

King Charles. Capo d'Istria, and Rovigno. Richard the Third, though he came in by blood, Trieste, a large and thriving sea-port of yet the short time of his triennial reign he was with Austria, the capital of a district in the Illyrian out any, and proved one of my best lawgivers, territory. It is situated near the north-west exHowel's England's Tears.

tremity of the gulf of Venice, and is divided TRIENNIAL PARLIAMENTS were established into the Old and New Town. The former stands at the revolution in 1688; but were abolished, on a hill, with a castle on the top; the New Town and the septennial parliaments enacted, upon the called also Theresienstadt, is on level ground, extinction of the rebellion in 1715.

intersected by a canal, and built with neatness TRIENS, in antiquity, a copper money of the and regularity. The population, at present about value of one-third of an as, which on one side 40,000, is on the increase. Trieste has good bore a Janus's head, and on the other a water-rat. streets, and a number of commodious buildings,

TRIENTALIS, chickweed winter-green, in but few that are large or striking, except the cathebotany, a genus of plants belonging to the class dral, the church that formerly belonged to the of heptandria, and order of monogynia; and in Jesuits, and the theatre: the cathedral is an the natural system ranging under the twentieth ancient, the theatre a modern building. This is order, rotaceæ. The calyx is heptaphyllous; almost the only sea-port for a very large tract of the corolla is equal and plane, and is divided the south of Germany, the Illyrian provinces, into seven_segments; the berry is unilocular and part of the Sclavonian; in short, for the and dry. There is only one species, viz. :—T. lon tract of Austrian territory extending from Europæa, which is indigenous, and the only Tyrol 10 Transylvania. Venice, though entitled genus of heptandria that is so. The stalk is since 1814 to all the privileges of an Austrian single, five or six inches high, terminated with sea-port, does not, from its distance, interfere with five, six, or seven oval pointed leaves ; from its trade; while Fiume is a small place, less adthe centre of which arise on long foot-stalks vantageously situated. Among the exports from commonly two white starry flowers, each gene- Trieste are the produce of the mines of Idria, sally consisting of seven oval and equal petals, and even of Hungary; linen, tobacco, woollens succeeded by a globular dry berry, covered with from different parts of the Austrian dominions ; a thin white rind, having one cell, and contain- also printed cotions from Switzerland. The iming several angular seeds.

ports consist of cotton, wool, hides, raisins, silks, TRI'ER. From try. One who tries experi- rice, oil from the Levant; wheat chiefly from mentally ; test.

Odessa; sugar, coffee, and other tropical products You were used

from the West Indies and Brasil. The trade of To say, extremity was the trier of spirits ;

the Adriatic is conducted in barks of twenty, That common chances common men could bear.

thirty, or forty tons: these and much larger ves

Shukspeare. sels enter with ease the inlet, in the form of a Courts of justice are bound to take notice of acts canal, which leads from the sea into the town, of parliament, and whether they are truly pleaded or and has on each side quays for vessels to load not; and therefore they are the triers of them. Hale. The ingenious triers of the German experiment

and unload. The harbour dues at Trieste are infound that their glass vessel was lighter when the

considerable. Each of the trading nations of air had been drawn out than before, by an ounce and Europe has a consul here. The quantity of very near a third.


goods conveyed by land to and from Trieste is There should be certain triers or examiners ap- very considerable; this conveyance is tedious, pointed by the state to inspect the genius of every but not expensive. Ship-building is carried particular boy.

Spectator. on with activity, and the sugar refining, the

making of white lead, soap, leather, paper, and

Those who are carried away with the spontaneous wax. At some distance from the town are salt- corrent of their own thoughts, must never humour

their minds in being thus triflingly busy. Locke. works, or pools for the reception of sea-water, which’in the summer months is evaporated by vine favor, the honours or aflictions of this life will

To a soul supported with an assurance of the dithe heat of the sun. Coal is obtained at a few iniles distance. Trieste is built near the site of be equally trifling and contemptible.

Triflers not ev'n in trifles can excel; Rogers. the Roman Tergeste; and there are some re

'Tis solid bodies only polish well.

Young mains of the aqueduct, partly subterranean, Brunetta's wise in actions great and rare, which brought water to it from a distance of But scorns on trifles to bestow her care : six miles. This town fell into the possession of Thus ev'ry hour Brunetta is to blame, Austria in 1382. In the fifteenth century Because the occasion is beneath her aim. it was a small place without trade; in 1719 it Think nought a trifle, though it small appear ; was made a free port by the Austrian govern- Small sands the mountain, moments make the year, ment; in 1753 the harbour was enlarged, and a

And trifles life. Your care to trifles give,

Id. mole formed to shelter it from the south: it is Or you may die before you truly live. open, however, to the Bora, an impetuonis north- renders the soul incapable of seeing, apprehending,

Whatever raises a levity of mind, a trifling spirit, east wind, which, did it not blow off the land, and relishing the doctrines of piety. Lat, would be dangerous to the shipping. The ter

TRIFOʻLIATE, adj. Lat. tres and folium. ritory belonging to the town comprises 170 square miles, and a population of nealy 9000. Ilaving three leaves. Trieste belonged to France during five years, Trifoliate cytisus restrained its boughs from 1809 to 1814. 212 miles S. S. W. of For humble sheep to crop, and goats to brouze.

Harte. Vienna, and sixty-nine E. N. E. of Venice.

TRI'FALLOW, v. a. Lat. tres and Sax. TRIFOLINUS, a mountain of Italy in Camfealga, a harrow. To plow land the third time pania, famous for its vines.- Plin. 14. c. 7. before sowing.

TRIFOLIUM, trefoil, or clover, in botany, a The beginning of August is the time of trifallow- genus of plants belonging to the class of diadeling, or last plowing before they sow their wheat. phia, and order of decandria, and in the natural

Mortimer. system ranging under the thirty-second order, TRIFISTULARY, adj. Lat. tres and fistula. papilionaceæ. The flowers are generally in Ilaving three pipes.

round heads; the pod is scarcely longer than Many of that species whose trifistulary bill

or crany the calyx, univalve, not opening, deciduous. we have bebeld.

Browne's Vulgar Errours. The leaves are three together. According to TRIFLE, v. n., v.a., & Belgic, tryfelen. Murray's edition of Linnæus there are forty-six TRI'FLER, n. s. (n. s. (To act or talk with- species; of which seventeen are natives of BriTRI'FLING, adj.

out weight or dig- tain. See Lightfoot's Flora Scotica, Berkenhout's TRI'FLINGLY, adv. nity; act or talk Synopsis, and Withering's Botanical Arrangewith levity or folly; mock; be of no importance: ments. The most remarkable are these :-1. T. to make unimportant (obsolete): a thing of no alpestre, long-leaved purple trefoil, or mountain moment: a trifler is one who acts with levity or clover, is thus characterised by Mr. Afzelius. folly: the adjective and adverb following cor The spikes are dense; the corollas somewhat respond with trifle.

equal; the stipulas are bristly and divergent; "Tis hard for every trifling debt of two shillings to

the leaflets lanceolated; the stalks stiff, straight, be driven to law.

Spenser. and very simple. It grows in dry, mountainous, When they say that we ought to abrogate such woody places, in Hungary, Austria, and Bohepopish ceremonies as are unprofitable, or else might mia, &c., but is not said by Mr. Afzelius to be a have other more profitable in their stead, they trifle native of Britain. 2. T. medium, according to and they beat the air about nothing which toucheth Mr. Afzelius, has been confounded with the


species pratense and alpestre; but it is to be Do not believe,

distinguished from them by having loose spikes, That, from the sense of all civility, I thus would play and trifle with your reverence.

corollas somewhat equal, stipulas subulate and Shakspeare.

connivent, and stalks flexuous and branched. The instruments of darkness tell us truths;

It is found in dry elevated situations, especially Win us with honest trifles, to betray us

among shrubs, or in woods where the soil is In deep consequence.

Id. Macbeth. chalky or clay, in England, Scotland, Sweden, Threescore and ten I can remember well,

Denmark, &c. 3. T. meliloti officinalis, the Within the volume of which time I've seen

melilot, has naked racemous pods, dispermous, Hours dreadful and things strange ; but this sore wrinkly, and acute, with an erect stalk. It night

grows in corn-fields and by the way sides, but Hath trifled former knowings.


not common. The stalk is erect, firm, striated, A man cannot tell whether Apollos or Albert branched, and two or three feet high; the Durer were the more triflers, whereon the one leaves ternate, smooth, obtusely oval, and serwould make a personage by geometrical proportions, rated; the flowers are small, yellow, pendulous, the other by taking the best parts out of divers faces and grow in long close spikes at the tops of the to make one excellent. Old Chaucer doth of Topas tell,

branches; the pod is very short, turgid, transMad Rabelais of Pantagruel,

versely wrinkled, pendulous, and contains either A later third of Dowsabell,

one or two seeds. The plant has a very pecuWith such poor trifles playing.

liar strong scent, and disagreeable bitter, acrid, Drayton's Nymphaid. taste, but such, however, as is not disagreeable


to cattle. The flowers are sweet-scented. It dus, or gray gurnard. 3. T. hirundo, the sap.. has generally been esteemed emollient and pharine gurnard. 4. T. lyra, or the piper. digestive, and been used in fomentations and TRIGLOCHIN, in botany, a genus of plants cataplasms, particularly in the plaster employed belonging to the class of hexandria, and order in dressing blisters; but is now laid aside, as its of trigynia; and in the natural system ranging quality is found to be rather acrid and irritating under the fifth order, tripelatoideæ. The calyx than emollient or resolvent. It communicates is triphyllous; the petals are three; there is no a most loathsome flavor to wheat and other style; the capsule opens at the base. There are grains, so as to render it unfit for making bread. three species; of which two are British, viz. :4. T. pratense, purple or red clover, is distin- 1. T. maritimum, or sea spiked grass, has ovate guished by dense spikes, unequal corollas, by sexlocular capsules; the stalk is short; the spike bearded stipulas, ascending stalks, and by the long, and flowers purplish. It is frequeni on calyx having four equal teeth. This is the the sea coasts. 2. T. palustre, arrow-headed botanical description of this species given by grass, has an oblong trilocular capsule. The Mr. Afzelius, who, in the Linnæan Transactions, stalk is simple, eight or ten inches high ; the vol. I., has been at much pains to remove three leaves long and narrow; the flowers are greenspecies of the trifolium from the confusion in ish, and grow at the end of a long spike. It is which they have been long involved ; namely, frequent in moist ground. Linnæus says that the pratense, medium, and alpestre. The red cattle eat these two species with avidity. clover is common in meadows and pastures, and TRI'GLYPH, n. s. In architecture. A memis the species which is generally cultivated as ber of the frize of the Dorick order set directly food for caule. It abounds in every part of over every pillar, and in certain spaces in the Europe, in North America, and even in Siberia. intercolurnniations. It delights most in rich, moist, and sunny places; The Dorick order has now and then a sober garyet Hourishes in dry, barren, and shady places. nishments of lions' heads in the cornice, and of tryFor an account of the mode of cultivating it, see gluphs and metopes always in the frize. Wotton. Rural Economy. 5. T. repens, white creeping trefoil, or Dutch clover, has a creeping stalk; A term in astrology.

TRIGON, n. s. Gr. Tpiywvov. A triangle. its power gathered into an umbellar head, and its pods tetraspermous. It is very common in

The ordinary height of a man ninety-six digits, fields and pastures. It is well known to be the ancient Egyptians estimated to be equal to that excellent fodder for cattle; and the leaves are a

mystical cubit among them stiled passus Ibidis, or good rustic hygrometer, as they are always ing of three latera, each thirty-two digits.

the trigon that the Ibis makes at every step, consistrelaxed and faccid in dry weather, but erect in

Hale's Origin of Mankind. moist or rainy:

A spar of a yellow hue shot into numerous trigonal TRIFORM, adj. Lat. triformis. Having a pointed shoots of various sizes, found growing to one triple shape.

side of a perpendicular fissure of a stratum of freeThe moon her monthly round

Woodward. Etill ending, still renewing through mid heaven, TRIGONELLA, fenugreek, in botany, a With borrowed light her countenance triform Hence fills, and empties, to enlighten the earth.

genus of plants belonging to the class of dia

delphia, and order of decandria; and in the Milton.

natural system arranged under the thirty-second TRIFURCATED (from tres and furca, a order, papilionacex. The vexillum and aleæ fork), having three prongs.

are nearly equal and patent, resembling a tripeTRIGA, in antiquity, a kind of car or chariot talous corolla. There are twelve species ; of drawn by three horses; whence the name. which the most remarkable is, T. fænum græcum, TRIGʻGER, n. s. Derived by Junius from

or fenugreek, a native of Montpelier in France. Fr. trigue, and Lat. intricare. See Tricker. A Fenugreek is an annual plant, which rises with catch to hold the wheel on steep ground. The a hollow, branching, herbaceous stalk, a foot and catch of a gun-lock.

a half long, garnished with trifoliate leaves, The pulling the trigger of the gun with which the placed alternately, whose lobes are oblong, oval, murder is committed, has no natural connection with indented on their edges, and have broad furrowthose ideas that make up the complex one, murder. ed foot-stalks. Fenugreek seeds have a strong


disagreeable smell, and an unctuous farinaceous TRIGINTALS, n. S. Lat. triginta. Thirty. taste, accompanied with a slight bitterishness.

Trentals or trigintals were a number of masses to The principal use of these seeds is in cataplasms the tale of thirty, instituted by Saint Gregory. and fomentations, for softening, maturating, and

Ayliffe. discussing tumors; and in emollient and carmiTRIGLA, in ichthyology, a genus of fishes native glysters. belonging to the order of thoracici. The head TRIGONOM'ETRY, n. s.

Gr. τριγωνος and is loricated with rough lines, and there are seven μετρο. .

See next page. rays in the membranes of the gills. There are Trigonometry is the art of measuring triangles, or eleven species, of which the principal are these: of calculating the sides of a triangle sought, and 1. T. cuculus, the red gurnard. 2. T. gurnar- this is plain or spherical.




TRIGONOMETRY is that branch of mathematical the angle, which they contain; by which is science by which, if certain parts of triangles meant merely that the intercepted arc is the same are given, the others may be computed.

part of the circumference of the circle that the Every triangle has six parts, three sides and angle is of four right angles. Thus, in fig. 1, three angles; and it is requisite that three of TriGONOMETRY, A B is the same part of the cir these be given, to find the other three. In sphe- cumference A B DF EA that the angle ACB rical trigonometry, the given parts may be of is of four right angles. any kind, either all sides or all angles, or part the 2. If the circumference of a circle be divided one, and part the other. But in plane trigono- into 360 equal parts, each of these parts is called metry, at least one of the given parts must be a a degree of the circle; if a degree be divided side; as from the angles alone only the proportions into sixty equal parts, each of these parts is callof the sides, not their actual lengths, can be de- ed a minute; and, if a minute be divided into termined.

sixty equal parts, each of these parts is called a The sides and angles of triangles being quan- second, &c.; and whatever number of degrees, tities of different kinds, they cannot be directly minutes, seconds, &c., are contained in any are compared with each other; but the relation be- of a circle, the angle at the centre measured by tween the sides and the magnitudes of the an that arc is said to contain the same number of gles may be found by comparing the sides with degrees, minutes, seconds, &c. certain lines drawn in and about a circle, on 3. Degrees, minutes, seconds, &c., are usually which lines the arcs of the circle which measure denoted by the marks ora" &c.; thus 18° 4' 27* the angles of the triangles depend. These lines signifies eighteen degrees, four minutes, and are called chords, sines, tangents, and secants. twenty-seven seconds. The ancients, Menelaus, Hipparchus, &c., per 4. Two arcs whose sum is equal to a semicirformed their trigonometrical computations by cle, or two angles whose sum is equal to two means of the chords; and the sines, as well as right angles, are called supplements of each the common theorems relating to them, were in- other. troduced into trigonometry by the Moors and 5. The difference between an arc and a quad. Arabians, from whom this art, with several other rant, or between an angle and a right angle, is branches of science, passed into Europe. Since called the complement of that arc or that angle. the fifteenth century, the Europeans have in 6. A perpendicular let fall from one extremity troduced the use of tangents, secants, &c., with of an arc upon the diameter which passes through the theorems relating to them.

the other extremity, is called the sine of that arc. Few circumstances have contributed more to 7. The versed sine of an arc is that portion the improvement this science than a simple of the diameter intercepted between the sine and suggestion respecting the notation, first made and the circumference. adopted in practice by Euler. It is nothing 8. The tangent of an arc is a perpendicular to more than denoting the angles of a triangle by the diameter at one extremity of an arc, meeting the first three capital letters of the alphabet, A, the diameter produced which passes through the B, and C, and the sides opposite those angles other extremity. by the corresponding smali letters a, b, and c; 9. The secant of an arc is the line drawn for in any theorem for the resolution of a prob- from the centre to the termination of the tanlem in trigonometry, the relation between the gent. parts is at once perceived.

10. The sine, tangent, secant, &c., of the comThus, in the common formulæ, cos. A = plement of an arc are usually termed the cosine, cos. Q - cos. b. cos. c

cotangent, cosecant, &c., of that arc, it is seen at once that a sin. b. sin. c

To illustrate the above definitions, let AB is the side opposite the angle A, and that b and (fig. 2) be the arc of a circle described with care the sides containing that angle.

the radius AC, and let A E be a quadrant; We have above given the original signification from B draw B D perpendicular to the diameter of the term trigonometry; but, in the modern A A'; and parallel to it draw AT, meeting CT acceptation of the term, it may be considered as in T; let G B and EM be drawn parallel to A the science by which we may determine the po- A', the latter meeting CT produced in M. Then sitions and dimensions of different parts of space, B A is the supplement of DEA', and B E A' is by means of the previous knowledge of some of the supplement of BA; B A is the complement those parts. The formulæ of trigonometry have of B E, and B E is the complement of B A or of also been applied to the solution of problems BE A', the angles B C A and B C A' are supplein which quantity, not magnitude, is the only ments of each other; and BCE is the compleconsideration; as in the solution of the irreduci- ment of B C A or of BCA.-BD is the sine, ble case of cubic equations; and of physical as DA the versed sine, AT the tangent, and CT tronomy, trigonomical formulæ may be said to the secant of the arc A B, or of the angle ACB form the language.

to the radius AC. GB is the sine, EG the Definitions.-1. If two lines meet in the cen versed sine, EM the langent, and C M the setre of a circle, the arc of the circumference in cant of the arc B; which arc being the con tercepted between them is called the measure of plement of A B, G B, or its equal C D, is called





cos. A B







the cosine, EG the coversed sine, EM the co

rad.: tangent, and CM the cosecant of the arc AB,

when radius is unity; cosect. or of the angle AC B to the radius A C. These expressions are thus abbreviated :

when radius is unity. For the sine of an arc as A B is put sin. A B

Again, AT:AC::CE:EM; or tan. : rad. cusine AB

:: rad. : cot. ; whence tan. · cot. = rado; tan. = tangent AB

tan, AB

rad.? cotangent AB

cot. A B

when radius is unity; cot. = cot. cot.

tan. secant AB

• sec. AB cosecant AB

cosec. A B, &c.

when radius is unity. From the preceding definition we may deduce 8. If M and N represent any two arcs, we the following obvious consequences :

have, from what has just been shown, cos. M · 1. When an arc vanishes, its sine and tangent sect. M = cos. N. sect. N; sin. M · cosect. M Fanish also; and its secant and cosine are each

= sin. N. cosect. N; and tan. Mocot. M. equal to the radius.

tan. N.cot. N. Whence cos. M : cos. N:: sect. 2. The sine and the versed sine of a quadrant N : sect. M; sin. M: sin. N :: cosect. N:coare each equal to the radius, its cosine vanishes, sect. M; and tan. M : tan. N : cot. N : cot. M. and its secant and tangent are infinite.

9. The sine, tangent, &c., of an arc which is 3. The versed sine of an arc and its cosine the measure of any given angle as A B C, fig. 3, are together equal to the radius.

&c. is to the sine of any other arc by which 4. The chord of an arc is twice the sine of the same angle A BC may be measured, as the half the arc.

radius of the first arc to the radius of the second. 5. Of any arc less than a quadrant, the arc is less For let A C and M N each measure the angle B; than its tangent, the chord less than the arc, and CD being the sine, D B the versed sine, A E the the sine less than the chord. For the sector

tangent, and B E the secant of the arc A C; NO CAB is less than the triangle CAT; and, by the sine, O M the versed sine, M P the tangent,


and B P the secant of the arc MN. Then by mensuration, the sector CAB

хагс 2

similar triangles we have CD:N():: rad. BC CA

: rad. BN;AE: M P or BE:BP:: rad. BA AB, and the triangle CAT=

: rad. BM; and BC:BD::BN:B0; or CA

BA:BD :: BM:B(); hence BA:BA

X A B is less than A T, and BD:: BM :BM — B 0; or BA: AD::B


M :M0; or BA:BM :: AD:MO. consequently A B is less than AT. In a similar 10. In trigonometrical investigations it is often way it may be shown that the chord A B is less convenient to consider the radius as unity, for than the arc A B, and the sine B D less than the the sake of simplifying the expressions. : But chord A B.

such expressions may easily be adapted to any 6. An arc and its supplement have the same other radius. For if BC in the figure under side, tangent, and secant.

consideration be represented by R, B N by unity, 7. The radius, tangent, and secant, constitute and DC = sin. to radius R, we have BC (R) : a right angled triangle=CAT the cosine, radius,


Hence, and sine, constitute a similar right angled trian- CD (sin.) :: BN (1):N O

R gle; as do also the cotangent, radius, and co any formula which has been investigated on the secant. Hence CD + BDP = C B'; or cos.? + sina supposition that radius is unity, may be adapted

to another radius R by substituting for sin., tan., -rad? ; CA+ AT = CT?; or rad.: + tan.?

sin. tan. = sec.?; C E? + EM’= CM?; or rad.? + cot.? &c., in the given expression,


&c., and = cosec.?; cos.' = rad.? -- sin.?= rad. + sin.

then reducing the expression to its most simple rad.- sin.; sin.? = rad.? — cos.' = rad. + cos. • form. rad

The numerical values of the sines, tangents, And, by similar triangles, we have CD:DB exhibit the ratio of the sines, tangents, &c., cor

&c., of every arc computated to any radius will :: CA:AT, or cos.: sin.:: rad. : tan., or tan. = rad. • sin. sin.

responding to any other radius. A table conwhen radius is unity. Simi- taining such numbers is called a table of natural

sines, tangents, &c.; and a table exhibiting the rad..cos. larly we have cot. =

when ra

logarithms of those numbers is called a table of sin.

logarithmic sines, tangents, &c. Tables of natudius is unity.

ral sines, &c., are generally computed to radius Again, CD:CB::CA:CT, or cos. : rad. unity; and tables of logarithmic sines, &c., are :: rad. : sect.; whence cos. sect. = rad.', or cos. =

generally computed to the radius whose logarithm rad.? 1

is 10, that the logarithm of the smallest sine when radius is unity; and sect. = likely to be required in computation, may have a rad.

positive, not a negative index; or that the corwhen radius is unity. Similarly we responding natural sine may not be fractional.

The logarithm of radius in such tables being

rad.? have sin. cosect. = rad.?; sin. =

10, the logarithm of rad.? is 20, of rad." 30, &c.,

the logarithmic sine and cosine of any arc is lesz VOL. XXII


whence CA

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