bly above the town, must have been once strong, but it now serves only as a jail. It is surmounted by the imperial flag, and there are placed upon it a few pieces of cannon for firing salutes.

Shipbuilding is carried on to a considerable extent. The chief manufactures are soap made of oil, pottery, white lead, leather, paper, majolica, vitriol, silk, rossoli, (of which 600,000 bottles are annually exported,) wax-bleaching, sugar-refining; and the manufacture of anchors, cordage, and sail cloth are also carried on.

The exports from Trieste comprehend the produce of the mines of Idria and even Hungary, tobacco, lime, Austrian woollens, and Swiss cottons. The imports are cotton wool, silks, hides, raisins, rice, oil from the Levant, wheat from Odessa, and colonial produce from the Brazils and the West Indies. The wine called Reinfall, from the vineyards of Prosek, is much esteemed. There are extensive salt works at Zaule and Servola. Coal is obtained a few miles from the town. The annual fair begins on the 1st and ends on the 24th of August. The territory belonging to the town occu pies 170 square miles, and contains a country population of nearly 9000.

About two leagues from the town is the interesting grotto of Corquale, so called from a village of that name. The road to it is over the summit of the mountain Paliso. The following account of it is given by Kuttner. "This grotto surpasses any that I ever beheld. The figures of the stalactites exhibit an uncommon variety of forms, and likewise a grander style and larger proportions than any I had ever yet met with. It is particularly distinguished for the columns on which the vaulted roof reposes, like that of a Gothic church. of these columns are 20, 30, or more feet in length, and of proportionable thickness. The flame of a torch, or of burning straw, produces a grand and picturesque effect. Many of the stalactites suspended from the roof are 12 or 15 feet in length, and at the top where they are united to it, are not less than 15 or 18 feet in circumference.

Many

"The grotto has the peculiarity that the entrance is not horizontal into a hill or eminence, but in a plain from which you are obliged to descend nearly in a perpendicular direction. You continue descending steep declivities, arriving now and then at nearly perpendicular shafts, in which a kind of stone steps have been cut; but they have been formed with so little care, and are partly rendered so slippery with water that is constantly dropping upon them, that you every moment run great risk of falling. We proceeded about a quarter of an hour when the steps ceased, and the perpendicular descent prevented our advancing any farther. I am informed that the length of this grotto has never been ascertained, but that, from various reasons, it is supposed to have a second opening at the distance of two German, upwards of nine English miles."

At the end of 1824, the population of Trieste and its territory, comprising the garrisons and

TRIGG, county of Kentucky, bounded by Caldwell N.W., Christian E., Montgomery county of Tennessee S. E., Stewart county of Tennessee S.W., and Tennessee river separating it from Calloway county, Kentucky, W. Length along the eastern boundary 34, mean breadth 12, and area 408 square miles. Extending in Lat. from 36° 38' to 37° 05' N., and in Lon. from 10° 42′ to 11° 12′ W. from W. C.

Cumberland river enters the southern boundary from the state of Tennessee, and flowing in a direction a little W. of N. divides Trigg into two unequal portions. The smaller section, a parallelogram between Tennessee and Cumberland rivers, contains about 80 square miles. The body of the county lies east of Cumberland river, and, with a westerly declivity, is drained by different branches of Little river, a small stream rising generally in Christian and Todd counties.

By the post office list of 1830, there were four post offices in this county, at Cadiz, the seat of justice, at Canton, Cerulean Springs, and at Lindsay's Mills.

Cadiz, the seat of justice, stands on Little river, near the centre of the county, about ninety miles N.W. by W. from Nashville in Tennessee, seventy-five S.E. by S. from Shawneetown, Gallatin county, Illinois, and by post road 226 miles S. W. by W. from W. C. N. Lat. 36° 45', and in Lon. 10° 48′ W. from W. C. Population of the county in 1820, 3874. See Synopsis, article UNITED STATES.

H 2

TRIGONOMETRY, (Tgirμergia from Tgroc a triangle, and μrgs I measure) in its original sense signified that part of mathematical science which treated of the admeasurement of triangles. Triangles were supposed to be either described upon a plane or upon the surface of a sphere, and hence the science was divided into plane trigonometry and spherical trigonometry.

Like every other department of science, the objects of trigonometry became more extended as knowledge advanced and discovery accumulated, and this department of mathematical science, which was at first confined to the solution of one general problem, viz. "given certain sides and angles of a triangle to determine the others," has now spread its uses over the whole of the immense domains of mathematical and physical science. In the wide range of modern analysis, there is scarcely a subject of investigation to which trigonometry has not imparted clearness and perspicuity by the use of its language and its principles; and the physical investigations of philosophers of our times are still more largely indebted for their conciseness, elegance, and generality, to the symbols and cǝtablished formulæ of this science.

In its present improved and enlarged state, trigonometry might not improperly be called the angular calculus; for however extensive and various its more remote uses and applications may be, its immediate object is to institute a system of symbols, and to establish principles by which angular magnitude may be submitted to computation, and numerically connected with other species of magnitude; so that angles and the quantities on which they depend, or which depend on them, may be united in the same analytical formulæ, and may have their mutual relations investigated by the same methods of computation that are applied to all other quantities.

It is not easy nor indeed possible to trace back trigonometry to its earliest origin. Most probably this science took its rise in the solution of some problems relative to heights or distances, so inaccessible as not to admit of direct measurement. In the rudest state of geometrical knowledge the similitude of equiangular triangles must have been known. This indeed is a property so obvious and striking, that the practical conviction of its truth must have long preceded its geometrical demonstration. This then being understood, it was an easy step to perceive that a small and measurable triangle might be constructed similar to a large and immeasurable one, so that the sides of the large one might consist of as many leagues as those of the small one of inches. Thus if the number of leagues in any one side of the large one be measured, and a line be drawn consisting of as many inches, a triangle constructed upon that line hav

ing the same angles as the large triangle, will have its other sides consisting of as many inches as there are leagues in the other sides of the large one. Such is in fact the foundation of the applicatign of trigonometry to the measurement of triangles.

In the first instance, trigonometry must have been considered not as a separate part of science, but simply as a class of geometrical problems. As however their number increased and their useful application became extended, they gradually assumed the form of a distinct science. The earliest work on this subject of which we have any record, is one by Hipparchus, who flourished about 150 years before the Christian era. Theon informs us that Hipparchus wrote a work on the Chords of Circular Arcs, a title from which we collect that the subject must have been trigonometry, or the doctrine of sines.

The earliest work extant on the subject is the spherics of Theodosius. In the first century Menelaus is said to have written nine books on trigonometry, of which, however, only three have come down. The earliest trigonometrical tables which we have received are those of Ptolemy, given in his Almagest. In these he adopts the sexagesimal division of the arc, whose chord is equal to the radius, and reckons all arcs by sixtieths of that arc, and all chords by sixtieths of the radius. In the eighth century sines or half-chords were introduced by the Arabians, which was the only striking improvement which the science received until the 15th century, when Purbach abandoned the sexagesimal division of the radius and constructed a table of sines to a radius of 600,000, computed for every ten minutes of the quadrant. Subsequently Regiomontanus adopted a radius of 1,000,000, and calculated sines for every minute.

In the sixteenth century several mathematicians contributed to the advancement of the science, particularly by improvements in the form of tables. By far the most conspicuous among these was the celebrated Vieta. This mathematician, in a work entitled "Canon Mathematicus seu ad triangula cum appendicibus," gave a table of sines, tangents, and secants for every minute of the quadrant to the radius of 100,000, with their differences. At the end of the quadrant the tangents and secants are extended to eight or nine places of figures. The second part of this work contains an account of the construction of tables, plane and spherical trigonometry, miscellaneous problems, &c.

The necessity of having accurate tables for all computations in physical and mathematical science, directed the attention of mathematicians at this period to this department of trigonometry in particular. Accordingly we find many elaborate computations, among which may be mentioned the

work of Rheticus, published subsequently by Otho his pupil, and afterwards corrected by Pitiscus.

This work contains a trigonometrical canon for every ten seconds of the quadrant to fifteen places of figures. The sines and cosines were computed for every second in the first and last degrees of the quadrant.

With all the assistance derivable from tables, the labour of trigonometrical computation was still excessive, and consequently the chances of error proportionally great. Stimulated by the desire to remove or overcome this difficulty, the celebrated Napier applied the energies of a powerful mind to the subject, and the result was the invention of logarithms. For the particulars of this admirable invention and its history, the reader is referred to our article LOGARITHMS.

To Napier we are also indebted for those technical methods of retaining a large class of formula, which are necessary for the solution of spherical triangles, and which are known by the name of Napier's rules and Napier's analogies. In the course of the following treatise, we shall have occasion to explain these.

Towards the end of the last century, trigonometry underwent a complete revolution, by being altogether transformed from a geometrical science, as it had heretofore been uniformly treated, into an analytical one. This change took its rise in that department of trigonometry called the theory of angular sections, and which was first successfully cultivated by Vieta. The object of this part of the science is to express the sines, cosines, tangents, &c. of any proposed multiples or submultiples of an arc, as functions of the arc itself, or of its sine, cosine, tangent, &c. This class of problems from its nature not admitting of geometrical investigation, the powers of analysis were necessarily resorted to, and its language and principles once admitted within the pale of trigonometry, spread their influence through the whole science, so that at length they reached its most elementary parts, and have now left nothing geometrical except the definitions, if indeed we can admit the necessity of giving even these a geometrical form.

To this happy subversion of the geometrical method and the substitution of the analytical, is due all that power and facility of investigation which the analyst receives from the formulæ of this science. To this is due the great generality of its theorems, the beautiful symmetry which reigns among the groups of results, the order with which they are developed one from another, offering themselves as unavoidable consequences of the method, and almost independent of the will or the skill of the author. The singular fitness with which the language of analysis adapts itself so as to represent, even to the eye, all this order and harmony, are effects too conspicuous not to be immediately perceived. Nor is the elegant form which the science has thus received from the hand of analysis, a mere object pleasurable to contemplate but barren of utility. All this order and

symmetry, which is given as well to the matter as to the form, as well to the things expressed as to the characters which express them, not only serves to impress the knowledge indelibly on the memory, but is the fruitful source of further improvement and discovery.

The advantages which have resulted from the conversion of trigonometry into an analytical science have not been, however, confined to trigonometry itself. Almost every part of physical science has felt the benefit of this change, and none in a greater degree than astronomy. Many of the most brilliant discoveries, with which modern times have enriched this science, could scarcely be expressed, much less discovered, without the aid of the language of analytical trigonometry.

One of the last improvements which trigonometry has received from analysis, is in the theory of angular sections before mentioned. Notwithstanding the attention which has been devoted to this subject by some of the most profound analysts from the time of Euler to the present day, it remained until a very late period in an imperfect state. Formulæ, expressing relations between the sine and cosine of an arc, and those of its multiples, were established by Euler, and subsequently confirmed by the searching analysis of Lagrange, which have since been proved inaccurate, or true only under particular conditions; and it was not until within a few years that the complete exposition of this theory was published, and general formula assigned, expressing those relations. In the year 1811, M. Poissin detected an error in a formula of Euler, expressing the relation between the power of the sine or cosines of an arc, and the sines and cosines of certain multiples of the same arc. But the most complete discussion of the subject, which has hitherto appeared in a separate form, is contained in a memoir read before the academy of sciences at Paris, by Poinsot, an eminent French mathematician, in the year 1823, and further developed by him in another memoir, published in the year 1825. The reader will also find some interesting discussions on this subject in the Bulletin Universel, scattered in various parts of that work for the last three years.

Works on trigonometry have been so numerous that it would be vain to attempt an enumeration of them here. The English elementary treatises have been, with two exceptions, uniformly geometrical. The first treatise in which the subject was presented in an analytical form was that of the late Professor Woodhouse of Cambridge. At a more recent period, a much more detailed treatise has been published by Dr. Lardner. This treatise is perhaps more exclusively analytical than any which has yet appeared. Dr. Lardner has borrowed from geometry no other principle except the proportionality of the sides of equiangular triangles. In this treatise we find a very complete analysis of angular sections, including all the recent corrections and improvements. The following treatise is abridged from this work by the consent of the author.

PLANE TRIGONOMETRY.

SECTION I.

Of Angles and Arcs.

(2.) The angular space which surrounds a point may be divided into four right angles by two straight lines, in directions perpendicular to each other, and passing through the point. If lines be supposed to be drawn through the point of intersection of these two lines, dividing each of the four right angles into ninety equal angles, each of these angles is called a degree, and, therefore, the entire angular space around the point consists of four times ninety, or 360 degrees.

If each of these angles, called degrees, be divided into 60 equal angles, each of these smaller subdivisions is called a minute.

In like manner, each minute being subdivided into sixty equal angles, these subdivisions are called seconds.

A second is the smallest angle which has received a distinct denomination. All smaller angles are usually expressed as decimal parts of a second. Degrees are expressed by O placed over their

number.

Thus, forty-five degrees, or half a right angle, is expressed 45°; thirty degrees, or a third of a right angle, thus, 30°.

Minutes are expressed by an accent' placed over their number, thus, 5' signifies five minutes; and seconds by a double accent", thus, 5" signifies five

seconds.

Thus,

35° 17' 10'5 signifies thirty-five degrees + 17 minutes + 10 seconds 5 tenths of a second.

(3.) In some foreign mathematical and physical works a different division of angles is used, with which it is necessary the student should be acquainted. It has been long considered that the division of the right angle into ninety equal parts

was unnatural and inconvenient, and several mathematicians, both British and continental, have from time to time proposed a decimal division. This has been actually carried into effect in France, and adopted by many writers of that country. They divide the angular space round a point into four hundred equal parts, which they call degrees. Each degree is divided into a hundred minutes,

and each minute into a hundred seconds, and so

The former division is called the sexagesimal, and the latter the decimal division.

(4.) It is an established principle of geometry, that the circumferences of different circles are proportional to their radii; and hence we infer that similar arcs of circles are also proportional to their radii, and vice versa. Two arcs of different cir

cles, therefore, which bear the same ratio to their respective radii must be similar, and therefore consist of the same number of degrees, minutes, and seconds.

From these principles it follows that an arc of one second of all circles is contained the same number of times in their radii, and from the calcu lation of the ratio of the circumference of a circle to its diameter, it is known that this number dif fers from 206265 by a small fraction. Therefore the radius of any circle differs from an arc of 206265 seconds by a small fractional part of a second. The circumference of a circle being incommensurable with its diameter, it is impossible to express the exact length of the radius in seconds and parts of a second.

(5.) Angles in plane geometry being in general those of triangles, are generally considered as not exceeding 180°. We shall, however, take a more general view of angular magnitude, and consider it like every other species of quantity as capable of unlimited increase as well as unlimited diminution. If a line be supposed to revolve round a given point, and in a given plane, it will, in one revolution, move through an angle of 360°; in one revolution and a quarter through 360°+90°. Or if 180° be called, the revolving radius in every revolution will move through the angle 2 and in every quarter of a revolution through, and in every half revolution through 7. In general if n be an integer. The radius, after a number of complete revolutions, will have moved through an angle expressed by 2n. If it has exceeded a complete number of revolutions by an angle, the angle which it has described, will be expressed by will be expressed by 2n7- If the angle it has 2n+, and if it fall short of a complete number, it described exceed an exact number of revolutions by half a revolution, we shall get its expression by changing into in the former formula, which gives 20+π = (2n+1)π.

In like manner, if the angle, which the revolving radius has moved through, exceed or fall short of a complete number of revolutions by a right angle, its expression will be found by changing into in each of the formula, which gives,

π

2

SECTION II.

Of Trigonometrical Terms and their Mutual Re

lations.

(6.) It is an established property of a right angled triangle, that if the ratio of any pair of its sides be known, the angles and the ratios of the other sides may be found. This forms the fundamental principle of trigonometry, the exponents of these ratios being here adopted as the criterions for the determination of the angles.

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 be the angle, y the opposite side, and a the containing side, and r the hypothenuse; the angle may be indifferently determined by any of the three numbers.

With the centre C, and the linear unit CA as radius, let a circle be described, and let another radius CA" be drawn, making any angle with the initial radius CA. From A" draw the perpendicu

lar A'P to the initial radius CA, and from A draw the tangent, and produce CA" to meet it at T. Now, if y, x, and r, be the three sides of a right angled triangle similar to A"PC, being the angle opposite to y, we have

y

A'P

g СА

=

a'

(8.) From the definitions of the trigonometrical terms which we have just given, their mutual rela tions become manifest. As these relations are in effect the fundamental principles of trigonometry, we shall here give them in some detail. By the definitions we have,

COS.@ -[4], cot.&= [4],

x

-[5], cosec.&=. -[6].

r

y

y

(9.) By squaring [1] and [4], and adding the results, we have

since r = x2+y3.

sin.+cos.∞ =

= 1,

tan.20+1=

y3+x2

72

... sec. 2=1+tan..

(15.) In like manner, by squaring [5] and adding unity, we obtain