981) { Females 659}1,435 Garrison, { at sea, bly above the town, must have been once strong, strangers, was 43,693 inhabitants. In 1826, it was Males 1,976 few pieces of cannon for firing salutes. Females 9955 Shipbuilding is carried on to a considerable ex Males ŞDifference 544. 776 Deaths tent. The chief manufactures are soap made of oil, pottery, white lead, leather, paper, majolica, Total population on 1st January 1826, 44,234 vitriol, silk, rossoli, (of which 600,000 bottles are Marriages in 1825, 410 annually exported,) wax-bleaching, sugar-refining; and the manufacture of anchors, cordage, and sail Repartition of this Population. cloth are also carried on. City of Trieste, 27,325 The exports from Trieste comprehend the pro Territory, 7,909 duce of the mines of Idria and even Hungary, to. , 8,000 bacco, lime, Austrian woollens, and Swiss cottons. 6,000 $ The imports are cotton wool, silks, hides, raisins, Strangers, 1,000 rice, oil from the Levant, wheat from Odessa, and colonial produce from the Brazils and the West In 44,234 dies. The wine called Reinfall, from the vineyards East Lon. 13° 47' 8". North Lat. 45° 38' 8''. of Prosek, is much esteemed. There are exten See Lascasas's Travels in Istria and Dalmatia, sive salt works at Zaule and Servola. Coal is ob- and Kuttner's Travels in Gerinany, foc. vol. iv. tained a few miles from the town. The annual fair begins on the 1st and ends on the 24th of Au. That part of Trieste along and adjacent to the gust. The territory belonging to the town occu. harbour is liable to very destructive inundations pies 170 square miles, and contains a country pop- from the wind tides, the only tides indeed that oculation of nearly 9000. cur to any serious amount in the Gulf of Venice. About two leagues from the town is the interest. The casual swell is, however, great, sudden, and ing grotto of Corquale, so called from a village of ruinous. Such a catastrophe occurred on the 8th that name. The road to it is over the summit of of October 1829, when the waves reached the under the mountain Paliso. The following account of it stories of all the lower part of the city, and prois given by Kuttner. “ This grotto surpasses any duced destructive consequences in cellars, storethat I ever beheld. The figures of the stalactites houses, warehouses and magazines. The loss was exhibit an uncommon variety of forms, and like immense. wise a grander style and larger proportions than TRIGG,county of Kentucky, bounded by Caldwell any I had ever yet met with. It is particularly N.W., Christian E., Montgomery county of Tendistinguished for the columns on which the vaulted nessee S. E., Stewart county of Tennessee S.W., and roof reposes, like that of a Gothic church. Many Tennessee river separating it from Calloway county, of these columns are 20, 30, or more feet in length, Kentucky, W. Length along the eastern boundary and of proportionable thickness. The flame of a 34, mean breadth 12, and area 408 square miles. torch, or of burning straw, produces a grand and Extending in Lat. from 36° 38' to 37° 05' N., and picturesque effect. Many of the stalactites sus in Lon. from 10° 42' to 11° 12' W. from W. C. pended from the roof are 12 or 15 feet in length, Cumberland river enters the southern boundary and at the top where they are united to it, are not from the state of Tennessee, and flowing in a diless than 15 or 18 feet in circumference. rection a little W. of N. divides Trigg into two “ The grotto has the peculiarity that the en unequal portions. The smaller section, a parallelotrance is not horizontal into a hill or eminence, but gram between Tennessee and Cumberland rivers, in a plain from which you are obliged to descend contains about 80 square miles. The body of the nearly in a perpendicular direction. You continue county lies east of Cumberland river, and, with a descending steep declivities, arriving now and then westerly declivity, is drained by different branches at nearly perpendicular shafts, in which a kind of of Little river, a small stream rising generally in stone steps have been cut; but they have been Christian and Todd counties. formed with so little care, and are partly rendered By the post office list of 1830, there were four so slippery with water that is constantly dropping post offices in this county, at Cadiz, the seat of upon them, that you every moment run great risk justice, at Canton, Cerulean Springs, and at Lindof falling. We proceeded about a quarter of an say's Mills, hour when the steps ceased, and the perpendicular Cadiz, the seat of justice, stands on Little river, descent prevented our advancing any farther. I near the centre of the county, about ninety miles am informed that the length of this grotto has N.W. by Wfrom Nashville in Tennessee, sevennever been ascertained, but that, from various rea ty-five S.E. by S. from Shawneetown, Gallatin sons, it is supposed to have a second opening at county, Illinois, and by post road 226 miles S. W. the distance of two German, upwards of nine Eng. by W. from W. C. N. Lat. 36° 45', and in Lon. lish miles.” 10° 48' W. from W. C. Population of the counAt the end of 1824, the population of Triestety in 1820, 3874. See Synopsis, article UNITED and its territory, comprising the garrisons and States. TRIGONOMETRY. one. TRIGONOMETRY, (T87Qvouritele from Tsywroc a tri- ing the same angles as the large triangle, will have angle, and uitgica I measure) in its original sense its other sides consisting of as many inches as signified that part of mathematical science which there are leagues in the other sides of the large treated of the admeasurement of triangles. Tri Such is in fact the foundation of the applicaangles were supposed to be either described upon tign of trigonometry to the measurement of triana plane or upon the surface of a sphere, and hence gles. the science was divided into plane trigonometry In the first instance, trigonometry must have and spherical trigonometry. been considered not as a separate part of science, Like every other department of science, the ob but simply as a class of geometrical problems. As jects of trigonometry became more extended as however their number increased and their useful knowledge advanced and discovery accumulated, application became extended, they gradually asand this department of mathematical science, sumed the form of a distinct science. The earliest which was at first confined to the solution of one work on this subject of which we have any record, general problem, viz. “ given certain sides and is one by Hipparchus, who flourished about 150 angles of a triangle to determine the others," has years before the Christian era. Theon informs us now spread its uses over the whole of the immense that Hipparchus wrote a work on the Chords of domains of mathematical and physical science. In Circular Arcs, a title from which we collect that the wide range of modern analysis, there is scarce the subject must have been trigonometry, or the ly a subject of investigation to which trigonometry doctrine of sines. has not imparted clearness and perspicuity by the The earliest work extant on the subject is the use of its language and its principles; and the physi- spherics of Theodosius. In the first century Menecal investigations of philosophers of our times are laus is said to have written nine books on trigostill more largely indebted for their conciseness, nometry, of which, however, only three have elegance, and generality, to the symbols and catab come down. The earliest trigonometrical tables lished formulæ of this science. which we have received are those of Ptolemy, In its present improved and enlarged state, trig- given in his Almagest. In these he adopts the onometry might not improperly be called the an sexagesimal division of the arc, whose chord is gular calculus; for however extensive and various equal to the radius, and reckons all arcs by sixtieths its more remote uses and applications may be, its of that arc, and all chords by sixtieths of the raimmediate object is to institute a system of sym. dius. In the eighth century sines or half-chords bols, and to establish principles by which angular were introduced by the Arabians, which was the magnitude may be submitted to computation, and only striking improvement which the science renumerically connected with other species of mag ceived until the 15th century, when Purbach abannitude; so that angles and the quantities on which doned the sexagesimal division of the radius and they depend, or which depend on them, may be constructed a table of sines to a radius of 600,000, united in the same analytical formulæ, and may computed for every ten minutes of the quadrant. have their mutual relations investigated by the Subsequently Regiomontanus adopted a radius of same methods of computation that are applied to 1,000,000, and calculated sines for every minute. all other quantities. In the sixteenth century several mathematicians It is not easy nor indeed possible to trace back contributed to the advancement of the science, partrigonometry to its earliest origin. Most proba- ticularly by improvements in the form of tables. bly this science took its rise in the solution of some By far the most conspicuous among these was the problems relative to heights or distances, so inac- celebrated Vieta. This mathematician, in a work cessible as not to admit of direct measurement. In entitled “ Canon Mathematicus seu ad triangula the rudest state of geometrical knowledge the sim- cum appendicibus,” gave a table of sines, tangents, ilitude of equiangular triangles must have been and secants for every minute of the quadrant to the known. This indeed is a property so obvious and radius of 100,000, with their differences. At the striking, that the practical conviction of its truth end of the quadrant the tangents and secants are must have long preceded its geometrical demon- extended to eight or nine places of figures. The stration. This then being understood, it was an second part of this work contains an account of the easy step to perceive that a small and measurable construction of tables, plane and spherical trigotriangle might be constructed similar to a large nometry, miscellaneous problems, &c. and immeasurable one, so that the sides of the The necessity of having accurate tables for all large one might consist of as many leagues as those computations in physical and mathematical sciof the small one of inches. Thus if the number of ence, directed the aitention of mathematicians at leagues in any one side of the large one be mea this period to this department of trigonometry in sured, and a line be drawn consisting of as many particular. Accordingly we find many elaborate inches, a triangle constructed upon that line hav- computations, among which may be mentioned the work of Rheticus, published subsequently by Otho symmetry, which is given as well to the matter as his pupil, and afterwards corrected by Pitiscus. to the form, as well to the things expressed as to This work contains a trigonometrical canon for the characters which express them, not only serves every ten seconds of the quadrant to fifteen places to impress the knowledge indelibly on the memory, of figures. The sines and cosines were computed but is the fruitful source of further improvement for every second in the first and last degrees of the and discovery. quadrant. The advantages which have resulted from the With all the assistance derivable from tables, the conversion of trigonometry into an analytical scilabour of trigonometrical computation was still ence have not been, however, confined to trigonomexcessive, and consequently the chances of error etry itself. Almost every part of physical science proportionally great. Stimulated by the desire to has felt the benefit of this change, and none in a remove or overcome this difficulty, the celebrated greater degree than astronomy:. Many of the most Napier applied the energies of a powerful mind to brilliant discoveries, with which modern times the subject, and the result was the invention of log- have enriched this science, could scarcely be exarithms. For the particulars of this admirable pressed, much less discovered, without the aid of invention and its history, the reader is referred to the language of analytical trigonometry. our article LOGARITHMS. One of the last improvements which trigonomeTo Napier we are also indebted for those techni. try has received from analysis, is in the theory of cal methods of retaining a large class of formula, angular sections before mentioned. Notwithstandwhich are necessary for the solution of spherical ing the attention which has been devoted to this triangles, and which are known by the name of subject by some of the most profound analysts from Napier's rules and Napier's analogies. In the course the time of Euler to the present day, it remained of the following treatise, we shall have occasion to until a very late period in an imperfect state. explain these. Formulæ, expressing relations between the sine Towards the end of the last century, trigonometry and cosine of an arc, and those of its multiples, underwent a complete revolution, by being altogether were established by Egler, and subsequently contransformed from a geometrical science, as it had firmed by the searching analysis of Lagrange, which heretofore been uniformly treated, into an analytical have since been proved inaccurate, or true only unone. This change took its rise in that department of der particular conditions; and it was not until trigonometry called the theory of angular sections, within a few years that the complete exposition of and which was first successfully cultivated by Vieta. this theory was published, and general formula asThe object of this part of the science is to express the signed, expressing those relations. In the year 1811, sines, cosines, tangents, &c. of any proposed mul. M. Poissin detected an error in a formula of Euler, tiples or submultiples of an arc, as functions of expressing the relation between the power of the the arc itself, or of its sine, cosine, tangent, &c. sine or cosines of an arc, and the sines and cosines This class of problems from its nature not admit of certain multiples of the same arc. But the ting of geometrical investigation, the powers of most complete discussion of the subject, which has analysis were necessarily resorted to, and its lan- hitherto appeared in a separate form, is contained guage and principles once admitted within the pale in a memoir read before the academy of sciences at of trigonometry, spread their influence through the Paris, by Poinsot, an eminent French mathematiwhole science, so that at length they reached its cian, in the year 1823, and further developed by most elementary parts, and have now left nothing him in another memoir, published in the year 1825. geometrical except the definitions, if indeed we The reader will also find some interesting discuscan admit the necessity of giving even these a geo sions on this subject in the Bulletin Universel, scatmetrical form. tered in various parts of that work for the last To this happy subversion of the geometrical me. three years. thod and the substitution of the analytical, is due Works on trigonometry have been so numerous all that power and facility of investigation which that it would be vain to attempt an enumeration of the analyst receives from the formulæ of this sci them here. The English elementary treatises have ence. To this is due the great generality of its been, with two exceptions, uniformly geometrical. theorems, the beautiful symmetry which reigns The first treatise in which the subject was present. among the groups of results, the order with which ed in an analytical form was that of the late Prothey are developed one from another, offering fessor Woodhouse of Cambridge. At a more rethemselves as unavoidable consequences of the me cent period, a much more detailed treatise has thod, and almost independent of the will or the been published by Dr. Lardner. This treatise is skill of the author. The singular fitness with The singular fitness with perhaps more exclusively analytical than any which which the language of analysis adapts itself so as has yet appeared. Dr. Lardner has borrowed from to represent, even to the eye, all this order and geometry no other principle except the proportionharmony, are effects too conspicuous not to be im- ality of the sides of equiangular triangles. In this mediately perceived. Nor is the elegant form treatise we find a very complete analysis of angular which the science has thus received from the hand sections, including all the recent corrections and of analysis, a mere object pleasurable to contem- improvements. The following treatise is abridged plate but barren of utility. All this order and from this work by the consent of the author. The former division is called the sexagesimal, PLANE TRIGONOMETRY. and the latter the decimal division. (4.) It is an established principle of geometry, SECTION I. that the circumferences of different circles are pro portional to their radii; and hence we infer that Of Angles and Arcs. similar arcs of circles are also proportional to their radii, and vice versa. Two arcs of different cir(2.), The angular space which surrounds a point cles, therefore, which bear the same ratio to their may be divided into four right angles by two respective radii must be similar, and therefore con. straight lines, in directions perpendicular to each sist of the same number of degrees, minutes, and other, and passing through the point. If lines be seconds. supposed to be drawn through the point of inter From these principles it follows that an arc of section of these two lines, dividing each of the four one second of all circles is contained the same right angles into ninety equal angles, each of these number of times in their radii, and from the calcu. angles is called a degree, and, therefore, the entire lation of the ratio of the circumference of a circle angular space around the point consists of four times to its diameter, it is known that this number difninety, or 360 degrees. fers from 206265 by a small fraction. Therefore If each of these angles, called degrees, be divided the radius of any circle differs from an arc of into 60 equal angles, each of these smaller subdivi. 206265 seconds by a small fractional part of a sesions is called a minute. cond. The circumference of a circle being inIn like manner, each minute being subdivided commensurable with its diameter, it is impossible into sixty equal angles, these subdivisions are called to express the exact length of the radius in seconds seconds. and parts of a second. A second is the smallest angle which has received (5.) Angles in plane geometry being in general a distinct denomination. All smaller angles are those of triangles, are generally considered as not usually expressed as decimal parts of a second. exceeding 180°. We shall, however, take a more Degrees are expressed by o placed over their general view of angular magnitude, and consider it number. like every other species of quantity as capable of Thus, forty-five degrees, or half a right angle, is unlimited increase as well as unlimited diminution. expressed 45°; thirty degrees, or a third of a right and in a given plane, it will, in one revolution, If a line be supposed to revolve round a given point, angle, thus, 30°. Minutes are expressed by an accent' placed over move through an angle of 360°; in one revolution their number, thus, 5' signifies five minutes; and and a quarter through 360° +90°. Or if 180° be seconds by a double accent", thus, 5" signifies five called 7, the revolving radius in every revolution seconds. will move through the angle 27 and in every quarThus, ter of a revolution through , and in every half 35° 17' 10"5 signifies thirty-five degrees + 17 minutes + 10 se revolution through 7. In general if n be an inteconds + 5 tenths of a second. ger. The radius, after a number of complete revo(3.) In some foreign mathematical and physical lutions, will have moved through an angle exworks a different division of angles is used, with pressed by 217. If it has exceeded a complete which it is necessary the student should be ac number of revolutions by an angle w, the angle quainted. It has been long considered that the which it has described, will be expressed by division of the right angle into ninety equal parts 2n7+-, and if it fall short of a complete number, it If the angle it has was unnatural and inconvenient, and several mathe- will be expressed by 2n7—2. maticians, both British and continental, have from described exceed an exact number of revolutions by time to time proposed a decimal division. This half a revolution, we shall get its expression by has been actually carried into effect in France, and changing - into - in the former formula, which adopted by many writers of that country. They gives 2n= + ==(2n+1)7. divide the angular space round a point into four In like manner, if the angle, which the revolvhundred equal parts, which they call degrees. ing radius has moved through, exceed or fall short Each degree is divided into a hundred minutes, gle, its expression will be found by changing « into and each minute into a hundred seconds, and so on. in each of the formulæ, which gives, Thus it is equally easy to express an angle in degrees and decimal parts of a degree, as in de 2n++* =(2n + b), and 2n=m * =(21—6) grees, minutes, and seconds. 369.567 329=36° 56' 73" 29"". The angle is called the complement of a, and The mark'" denoting hundredth parts of a second. the angle is called the supplement of c. The degree may here, therefore, be taken as the angular unit. 2 2 SECTION II. lations. As there are three pairs of sides in a right angled triangle differently related to either of its acute angles, so there are three ratios which will determine the angle. Let a be the angle, y the opposite side, and x the containing side, and r the hypothenuse; the angle c may be indifferently determined by any of the three numbers. y y r y 2 sin.co sin.a COS. COS. (8.) From the definitions of the trigonometrical The first is called the sine of the angle the terms which we have just given, their mutual rela. tions become manifest. As these relations are in y effect the fundamental principles of trigonometry, second is called its tangent, and the third - is we shall here give them in some detail. By the definitions we have, sin.c=\ [1], * [ [3], The three ratios which are the reciprocals of those already expressed, scil. COS.c= = [4], y (9.) By squaring [1] and [4], and adding the rebear the same relation to the other acute angle of sults, we have the triangle as the former do to the assumed one. sin."w+cos. Ww=1, One acute angle being the complement (5.) of the since y = x++yo. other, it follows that and are the sine, (10.) By dividing [1] by [4], we have y = tan.a, (11.) By multiplying [2] and [5], we obtain tangent, and co-secant. tan.o cot.w=l. (7.) We shall now explain the origin of these Thus the tangent and cotangent are reciprocals. denominations. With the centre C, and the linear unit CA as (12.) In like manner, by multiplying [3] and [4], radius, let a circle be described, and let another ra we have dius CA" be drawn, making any angle w with the sec.w cos.w=). initial radius CA. From A" draw the perpendicu. The secant and cosine are therefore reciprocals. lar A'P to the initial radius CA, and from A draw (13.) It follows also by multiplying [1] and [6], the tangent, and produce CA" to meet it at T. that Now, if y, x, and r, be the three sides of a right cosec.c sin.c=1, angled triangle similar to A'PC, co being the angle therefore the cosecant and sine are reciprocals. opposite to y, we have y (14.) By squaring [2], and adding unity to the result, we find CA p? But since CA is assumed as the linear unit, ... A"P=sin. . X2 Again, since TCA is also similar to the same ... sec. *-=1+tan. • triangle, we have (15.) In like manner, by squaring [5] and adding y TA unity, we obtain CA cosec. *w =l+ cot.*. (16.) By [9.] and [10.] it follows that sin.ee sin. a. tan.*+1=yo+x tan.w COS. |