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tween the segments of the base, and each of the sides about the .right angle is a mean proportional between the base and the adjacent segment.

There are also several curious problems in the Diophantine Analysis relating to right-angled triangles; viz.

1. The number which expresses

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the area of a rational right-angled —b) (4s—c)}

triangle cannot be equal to a square number.

2. Let b represent the base of a right-angled triangle, p the perpendicular, and the hypothenuse; then if we assume

b= r2-s2, and p = 2rs, we shall have

p = r2+s2;

that is, the triangle will be a rational one, rand's being assumed at pleasure.

or

3. Or if in the fraction

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the numerator and de

2rs 9 nominator be taken for the sides of a right-angled triangle, it will be a rational one. And as we may here give any values to r and s at pleasure, we have, in the first instance, by assuming rs+ 1, and taking successively

S=1, 2, 3, 4, 5, &c. the following remarkable series; viz.

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25+ 1

11, 23, 38, 4, 511, &c.

=

each of which terms, reduced to an improper fraction, will give the sides of a rational right-angled triangle.

4. And in the second form, by making s=1, and r=n+2, it be

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4. Log. of area➡

log.is +

log. (}s—a) + log. (§ s − b) +log.

(ts—c)}

Spherical TRIANGLES. A spheri cal triangle is that which is form ed by the intersecting arcs of three great circles of a sphere. Spherical triangles are divided into rightangled, oblique, equilateral, isosceles, scalene, &c. the same as plane triangles, and in the same cases, and therefore require no separate definition. They are also said to be quadrantal, when they have one side a quadrant.

Two sides, or two angles, are said to be of the same affection, when they are at the same time either both greater or both less than a quadrant, or 90°; and of different affections, when one is greater than a quadrant, and one less.

Spherical TRIANGLES have several useful properties, of which the following are the most important.

1. In spherical triangles, as well as in plane triangles, equal angles are subtended by equal sides; the greater angle is subtended by the greater side; and the less angle by the less side.

2. Any side or angle of a spherical triangle is less than a semicircle, or 180°.

3. The sum of any two sides of a spherical triangle is greater than the third side; and their difference and assum- is less than the third side.

n=1, 2, 3, 4, &c. we have this other series, 13, 211, 318, 418, 53, &c. which has the same property the former.

as

To find the Area of a Triangle. Let a, b, c, represent the three sides of the triangle; A, B, C, the an

532

4. The difference of any two sides of a spherical triangle is less than a semicircle, or 180°; and the sum of the three sides is less than 360°.

5. The sum of the three angles of than two right angles, or 180°; and any spherical triangle, is greater less than six right angles, or 540°. 6. The sum of any two angles of

a spherical triangle is greater than the supplement of the third angle. 7. If the three sides of a spherical triangle be equal to each other, the three angles will also be equal; and vice versa.

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 versa.

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 versâ.

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.

of table of numbers, arranged in the form of a triangle, formerly employed in arithmetical calcula tions; but it is now rendered useless, by the numerous improve. ments 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 instrument 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.

TRIGONOMETRY, the art of measuring the sides and angles of triangles, either plane or spheri cal, 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 practised 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 quanti ty of an angle is estimated by the number of degrees, minutes, &c. contained in the arc by which it is bounded; the degrees being de The area of any spherical trian-uoted by a small, the minutes by gle, 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.

a dash, the seconds by two dashes, and so on; thus 70 degrees, 16 minutes, 34 seconds, are denoted by 70° 16 34".

The complement of an arc or angle is what it wants of 90o, 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 Arithmetical TRIANGLE, is a kind | 40°, and its supplement 130°.

Solution of the Cases of Plane | fig. 1,) the sine of the right-angle 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 trigononietry.

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

Therefore, 1st, in any right-angled triangle BAC (Plate Trigonometry,

A, or radius, is to the hypothenuse
BC: sin. C. AB sin. B: AC.

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 CA C: AB, we may always substitute sin. B: cos. BAC: A B.

But since sin. B cos B= tang. B: R, we have AC: AB tang. B: R cot. C: R

These principles suffice to solve all the cases of the right-angled triangle ABC, when besides the right-angle A, we know any two of the five parts B, C, A B, AC, BC; 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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