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a spherical triangle is greater than the supplement of the third angle. 7. #5. three sides of a spherical triangle be equal to each other, the three angles will also be equal; and vice versä. 8. If any two sides of a spherical triangle be equal to each other, their opposite angles will also be equal; and vice versa. 9. If the sum of any two sides of a spherical triangle be equal to 180° ; the sum of their opposite angles will also be equal to 180°; and vice versâ. 10. If the three angles of a spherical triangle be all acute, or all right, or all obtuse, the three sides will be accordingly all less than 90°, or all equal to 90°, or all greater than 90° ; and vice versa. 11. Half the sum of any two sides of a spherical triangle, is of the same kind as half the sum of their opposite angles; or the sum of any two sides is of the same kind, with respect to 180°, as the sum of their opposite angles. To these may also be added the following properties of the polar triangle, by which the data in any case may be changed from sides to angles, and from angles to sides. If the three arcs of great circles be described from the angular points of any spherical triangle as poles, or at 90° distance from them, the sides and angles of the new triangle so formed, will be the supplements of the opposite angles and sides of the other; and vice versa. To find the Area of a Spherical Triangle.—The area, or surface, of a spherical triangle, is equal to the difference between the sum of its three angles, and two right angles, multiplied by the radius of the sphere. The area of any spherical triangle, which is small with regard to the whole sphere, (such as those used in geodetical operations) is very nearly equal to the area of a rectilinear triangle, of which the sides are equal in length to the former, and whose angles are each less than the corresponding angles of the former, by one-third of the spherical excess. Arithmetical TRIANGLE, is a kind

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of table of numbers, arranged in the form of a triangle, formerly employed in arithmetical calculations; but it is now rendered useless, by the numerous improvements of modern times, and therefore any minute description of it seems useless.

TRIANGULAR Compasses, are those which have three legs, by means of which a triangle may be taken off at once, which renders them

very useful in the construction of

maps, charts, &c. TRIBOMETER, the name given by Musschenbroek to an insurument invented by him for measuring the friction of metals. TRIDENT, a term used by Des Cartes for a particular kind of parabola employed by him for the construction of equations of the sixth degree. TRIGO NOM ETRY, the art of measuring the sides and angles of triangles, either plane or spherical, or rather of determining certain parts which are unknown from others that are given. This is a science of the greatest importance in almost every branch of mathematics, and particularly in astronomy, navigation, surveying, dialling, &c. and has been accordingly Pota by mathematicians of all ages. Trigonometry is called plane and spherical, according to the species of triangles to which it is applied. In plane trigonometry every circle is supposed to be divided into 360 equal parts, or degrees, every degree into 60 minutes, and each minute into 60 seconds, and so on ; and the measure or quantity of an angle is estimated by the number of degrees, minutes, &c. contained in the arc by which it is bounded ; the degrees being denoted by a small “; the minutes by a dash, the seconds by two dashes, and so on ; thus 70 degrees, 16 mi. nutes, 34 seconds, are denoted by 70° 16' 34/1. The complement of an arc or angle is what it wants of 900, or of a quadrant. And the supplement of an angle is what it wants of a semicircle or of 180°. Thus, if an angle measures 50°, its complement is 40°, and its supplement 130°. 2 Y3


Sotation of the Cases of Plane Triangles.

Any right-lined triangle may be inscribed in a circle; and this being done, each side of the triangle will be the chord of an arc double of that which measures the opposite angle ; that is, double the sine of that angle measured in the circle; therefore the sides of the triangle are to each other as the sines of the opposite angles measured in the same circle, and consequently as the sines of the same angles measured in the circle whose radius is that of the tables. Hence the following proposition of such frequent use in the practice of trigonometry.

In any triangle the sines of the angles are proportional to the sides opposite those same angles.

herefore, 1st, in any right-angled

triangle BAC (Plate Trigonometry,

fig. 1,) the sine of the right-angle A, or radius, is to the hypothen use B C = sin. C. A B = sin. B : A C. Second : Since in all right-angled triangles the sine of one acute angle C, is the cosine of the other acute angle B, we have sin. C = cos, B and reciprocally. Therefore, instead of the proportion sin. B : sin C = A C : A B, we may always substitute sin. B : cos. B = A C : A B. But since sin; B cos B = tang. B. : R, we have A C : A B tang. B : R = cot. C : R These principles suffice to solve all the cases of the right-angled triangle A B C, when besides the right-angle A, we know any two of the five parts B, C, AB, AC, B C ; provided they are not the two an: gles. In this latter case we can only determine the ratios of the three sides.

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