TRIGONOMETRY

TRIGONOMETRY,

Plane. THE art of measuring the sides and angles of triangles, either plane or spherical, whence it is accordingly called either PLANE TRIGONOMETRY, or SPHERICAL TRIGONOMETRY.

Trigonometry is an art of the greatest use in the mathematical sciences, especially in astronomy, navigation, surveying, dialing, geography, &c. &c. By it we come to know the magnitude of the earth, the planets and stars, their distances, motions, eclipses, and almost all other useful arts and sciences. Accordingly we find this art has been cultivated from the earliest ages of mathematical knowledge.

Trigonometry, or the resolution of triangles, is founded on the mutual proportions which subsist between the sides and angles of triangles; which proportions are known by finding the relations between the radius of a circle and certain other lines drawn in and about the circle, called cords, sines, tangents, and secants. The ancients, Menelaus, Hipparchus, Ptolemy, &c. performed their trigonometry by means of the cords. As to the sines, and the common theorems relating to them, they were introduced into trigonometry by the Moors or Arabians, from whom this art passed into Europe, with several other branches of science. The Europeans have introduced, since the 15th century, the tangents and secants, with the theorems relating to them.

The proportion of the sines, tangents, &c. to their radius, is sometimes expressed in common or natural numbers, which constitute what we call the tables of natural sines, tangents, and secants. Sometimes it is expressed in logarithms, being the logarithms of the said natural sines, tangents, &c.; and these constitute the table of artificial sines, &c. Lastly, sometimes the proportion is not expressed in numbers; but the several sines, tangents, &c. are actually laid down upon lines of scales; whence the line of sines, of tangents, &c.

In trigonometry, as angles are measured by arcs of a circle described about the angular point, so the whole circumference of the circle is divided into a great number of parts; as 360 degrees, and each degree into 60 minutes, and each minute into 60 seconds, &c.; and then any angle is said to consist of so many degrees, minutes, and seconds, as are contained in the arc that measures the angle, or that is intercepted between the legs or sides of the angle.

Now the sine, tangent, and secant, &c. of every degree and minute, &c. of a quadrant, are calculated to the radius $r$, and ranged in tables for use; as also the logarithms of the same; forming the triangular canon. And these numbers, so arranged in tables, form every species of right-angled triangles; so that no such triangle can be proposed, but one similar to it may be there found, by comparison with which the proposed one may be computed by analogy or proportion.

PLANE TRIGONOMETRY.

THERE are usually three methods of resolving triangles, or the cases of trigonometry; viz. geometrical construction, arithmetical computation, and instrumental operation. In the 1st method, the triangle in question is constructed by drawing and laying down the several parts of their magnitudes given, viz. the sides from a scale of equal parts, and the angles from a scale of cords or other instrument; then the unknown parts are measured by the same scales, and so they become known.

In the 2d method, having stated the terms of the proportion according to rule, which terms consist partly of the

numbers of the given sides, and partly of the sines, &c. of angles taken from the tables, the proportion is then resolved like all other proportions, in which a 4th term is to be found from three given terms, by multiplying the 2d and 3d together, and dividing the product by the 1st. Or, in working with the logarithms, adding the logarithm of the 2d and 3d terms together, and from the sum subtracting the logarithm of the 1st term; then the number answering to the remainder is the 4th term sought.

To work a case instrumentally, as suppose by the logarithm lines on one side of the two foot scales: Extend the compasses from the 1st term to the 2d or 3d, which happens to be of the same kind with it; then that extent will reach from the other term to the 4th. In this operation, for the sides of triangles, is used the line of numbers (marked Num.); and for the angles, the line of sines or tangents (marked sin. and tan.) according as the proportion respects sines or tangents. See SECTOR.

In every case of plane triangles there must be three parts, one at least of which must be a side. And then the different circumstances, as to the three parts that may be given, admit of three cases or varieties only; viz.

1st, When two of the three parts given are a side and its opposite angle. 2d, When there are given two sides and their contained angle. 3d, And, thirdly, when the three sides are given.

To each of these cases there is a particular rule or proportion adapted for resolving it by.

1st, The Rule for the 1st Case, or that in which, of the three parts that are given, an angle and its opposite side are two of them, is this, viz. that the sides are proportional to the sines of their opposite angles; that is,

As one side given
To the sine of its opposite angle : :
So is another side given
To the sine of its opposite angle.

Or,

As the sine of an angle given : :
To its opposite side : :
So is the sine of another angle given : :
To its opposite side.

So that, to find an angle, we must begin the proportion with a given side that is opposite to a given angle; and to find a side, we must begin with an angle opposite to a given side.

Example. Suppose in the triangle BDC (fig. 1.) there be plate DXTI. given the side $BC\; =\; 106$, $DB\; =\; 65$, and the angle $BCD\; =\; 31^\{\backslash circ\}\; 49\text{'}$ given; to find the angle BDC obtuse and the side CD.

1. Geometrically by Construction.

Draw the line BC equal to 106, at C make an angle of $31^\{\backslash circ\}\; 49\text{'}$ by drawing CD, take 65 in your compasses, and with one foot in B lay the other upon the line CD in D; draw the line BD, and it is done; for the angle D will be $120^\{\backslash circ\}\; 43\text{'}$, the angle B $27^\{\backslash circ\}\; 28\text{'}$, and the side DC 56.9 as was required.

2. Arithmetically by Logarithms.

As the side BD 65	log.	7.81291
Is to sine angle C $31^\{\backslash circ\}\; 49\text{'}$		9.72198
So is the side BC 106		2.02534
		11.74729
		1.81291
To sine angle D $120^\{\backslash circ\}\; 43\text{'}$		9.93438
		To

Plane.	To find DC.	180.0
As fine ang. C $31^\{\backslash circ\}\; 49\text{'}$	9.72198	The supp. 59.17 of ang. D.
Is to the side BC 65	1.91291	120.43 angle D.
So is fine ang. B $27^\{\backslash circ\}\; 28\text{'}$	9.66392	31.49 angle C.
	11.47683	152.32 their sum.
	9.72198	180.0
To the side DC 56.88	1.75485	152.32 sum subt.
		27.28 angle B.

Here it may be proper to observe, that if the given angle be obtuse, the angle sought will be acute; but when the given angle is acute, and opposite to a lesser given side, then the required angle is doubtful, whether acute or obtuse; it ought therefore to be determined before the operation. For it is plain the above proportion produces $59^\{\backslash circ\}\; 17\text{'}$ for the required angle; but as it is obtuse, its supplement to 180 degrees must be taken, viz. $120^\{\backslash circ\}\; 43\text{'}$.

By Gunter.

"The extent from 65 to 106 on the line of numbers will reach from $31^\{\backslash circ\}\; 49\text{'}$ to $59^\{\backslash circ\}\; 17\text{'}$ on the line of sines."

2dly, "The extent from $31^\{\backslash circ\}\; 49\text{'}$ to $27^\{\backslash circ\}\; 28\text{'}$ on the line of sines will reach from 65 to 56.88 on the line of numbers."

CASE II. When there are given two sides and their contained angle, to find the rest, the rule is this:

As the sum of the two given sides :

Is to the difference of the sides :

So is the tangent of half the sum of the two opposite angles or cotangent of half the given angle :

To tang. of half the diff. of those angles.

Then the half diff. added to the half sum, gives the greater of the two unknown angles; and subtracted leaves the less of the two angles.

Hence, the angles being now all known, the remaining 3d side will be found by the former case.

Example. The side BC = 109, BD = 76 (fig. 2.), and the angle CBD $101^\{\backslash circ\}\; 30\text{'}$ given, to find the angle BDC or BCD, and the side CD.

1. Geometrically by Construction.

Draw the line BC 109, and BD, so as to make an angle with BC of $101^\{\backslash circ\}\; 30\text{'}$, and make BD equal to 76; join BC and BD with a right line, and it is done; for the angle D being measured by the cord of $60^\{\backslash circ\}$, will be $47^\{\backslash circ\}\; 32\text{'}$, angle C $30^\{\backslash circ\}\; 58\text{'}$, and the side DC 144.8, as was required.

2. Arithmetically by Logarithms.

Side BC	109	-	109	-	180o 0'
BD	76	-	76	-	101 30
Their sum	185		33 their diff.	78 30	sum of the ang.
					D and C.

$\backslash frac\{1\}\{2\}$ Sum 39 15 then

To find the angles D and C.

As the sum of the sides BC and BD = 185	2.26717
Is to their difference	33 1.51851
So is tang. of $\backslash frac\{1\}\{2\}$ the sum of the angles C and D $39^\{\backslash circ\}\; 15\text{'}$	9.91224
	11.43075
	2.26717

To the tang of $\backslash frac\{1\}\{2\}$ the diff. of the angles C and D $8^\{\backslash circ\}\; 17\text{'}$ 9.16358

To half the sum of the angles D and C	39o 15
Add half the difference of the angles C and D	8 17

Gives the greater angle D 47 32
Subtracted, gives the lesser angle C 30 58

Plane.	To find DC.	180.0
As fine angle D $47^\{\backslash circ\}\; 32\text{'}$	-	9.86786
Is to the side BC 109	-	2.03743
So is fine angle B $101^\{\backslash circ\}\; 30\text{'}$	-	9.99119
		12.02862
		9.86786
To the side DC required 144.8	-	2.16076

3. By Gunter.

1st, "The extent from 185 to 33 on the line of numbers will reach from $39^\{\backslash circ\}\; 15\text{'}$ to $8^\{\backslash circ\}\; 17\text{'}$ on the line of tangents. 2dly, "The extent from angle D $47^\{\backslash circ\}\; 32\text{'}$ to $78^\{\backslash circ\}\; 30\text{'}$ (the supplement of angle B) on the line of sines, will reach from the side BC 109 to 144.8, the side DC required, on the line of numbers."

CASE III. Is when the three sides are given, to find the three angles; and the method of resolving this case is, to let a perpendicular fall from the greatest angle upon the opposite side or base, dividing it into two segments, and the whole triangle into two smaller right-angled triangles: then it will be,

As the base or sum of the two segments :

Is to the sum of the other two sides :

So is the difference of those sides :

To the difference of the segments of the base.

Then half this difference of the two segments added to the half sum, or half the base, gives the greater segment, and subtracted gives the less. Hence, in each of the two right-angled triangles, there are given the hypotenuse, and the base, besides the right angle, to find the other angles by the first case.

Example. The sides BC (fig. 3.) = 105, BD = 85, and CD = 50, given to find the angles BDC, BCD, or CBD.

1. Geometrically by Construction.

Draw the line BC equal to 105, take CD 50 in your compasses, and with one foot in C describe an arch; then take BD 85 in your compasses, and with one foot in B cut the former arch in D, join BD and DC, and it is done; for the angle B, being measured, will be found $28^\{\backslash circ\}\; 4\text{'}$, angle C $53^\{\backslash circ\}\; 7\text{'}$, which being added together, is $81^\{\backslash circ\}\; 11\text{'}$, their sum subtracted from 180, leaves angle D $98^\{\backslash circ\}\; 49\text{'}$ as was required.

2. Arithmetically by Logarithms.

The two shortest sides are BD (= 85) and CD (= 50), the sum of which is 135, and their difference 35. The segments of the base BC are found in this manner:

As the side BC	= 105	log. 2.02119
Is to the sum of the sides BD & DC	= 135	2.13033
So is their difference	= 35	1.54407
To the difference of the seg. of BC	= 45	1.65321

Thus the sum and difference of the segments of the base BC being known, we have only to add half this sum = $52\backslash frac\{1\}\{2\}$ to half the difference = $22\backslash frac\{1\}\{2\}$, and we shall obtain the greater segment, which is = 75; which subtracted from 105, gives 30 = the smaller segment. Then

To find the angle BDA.

As the hypotenuse BD	= 85	log. 1.92942
Is to radius	=	10.00000
So is the greater segment	= 75	1.87506
To the sum of the angle BDA	=	9.94504

The angle BDA therefore is equal to $61^\{\backslash circ\}\; 56\text{'}$

Let us now find the angle ADC, which is done thus.
As the hypotenuse DC = 50 log. 1.69897
Is to radius = 10.00000
So is the smaller segment = 30 1.47712
To the fine of ADC = 9.77315
The angle ADC therefore is equal to $36^\{\backslash circ\}\; 53\text{'}$, and the whole angle BDC = $98^\{\backslash circ\}\; 49\text{'}$.

To find the angle at B, we have only to subtract the angle BDA ($=\; 61^\{\backslash circ\}\; 56\text{'}$) from $90^\{\backslash circ\}$, and the rem. $28^\{\backslash circ\}\; 4\text{'}$ is the angle sought. The angle at C is equal to $53^\{\backslash circ\}\; 7\text{'}$.

3. By Gunter.

1st, 'The extent from 105 to 135, will reach from 35 to 45 on the line of numbers.' 2dly, 'The extent from 85 to 75, on the line of numbers, will reach from radius to $61^\{\backslash circ\}\; 56\text{'}$, the angle BDA on the line of fines.' 3dly, 'The extent from 50 to 30 on the line of numbers, will reach from radius to angle ADC $36^\{\backslash circ\}\; 53\text{'}$ on the line of fines.'

The foregoing three cases include all the varieties of plane triangles that can happen, both of right and oblique-angled triangles. But besides these, there are some other theorems that are useful upon many occasions, or suited to some particular forms of triangles, which are often more expeditious in use than the foregoing general ones; one of which, for right-angled triangles, as the case for which it serves so often occurs, may be here inserted, and is as follows.

CASE IV. When, in a right-angled triangle, there are given the angles and one leg, to find the other leg, or the hypotenuse. Then it will,

• As radius :
• To given leg AB ::
• So tang. adjacent the angle A :
• To the opposite leg BC, and ::
• So sec. of same angle A :
• To hypot. AC :

Example. In the triangle ABC (fig. 4.), right-angled at B,

Given the leg AB = 162
and the angle A = $53^\{\backslash circ\}\; 7\text{'}\; 48\text{'}\text{'}$
conseq. the angle C = $36^\{\backslash circ\}\; 52\text{'}\; 12\text{'}\text{'}$ } to find BC
and AC.

1. Geometrically.—Draw the leg AB = 162 : Erect the indefinite perpendicular BC : Make the angle A = $53^\{\backslash circ\}\; 7\text{'}\; 48\text{'}\text{'}$, and the side AC will cut BC in C, and form the triangle ABC. Then, by measuring, there will be found AC = 270, and BC = 216.

2. Arithmetically.

As radius	= 10	-	-	log. 10.0000000
To AB	= 162	-	-	2.2095150
So tang. A	= $53^\{\backslash circ\}\; 7\text{'}\; 48\text{'}\text{'}$	-	-	10.249372
To BC	= 216	-	-	2.3344522
So sec. A	= $53^\{\backslash circ\}\; 7\text{'}\; 48\text{'}\text{'}$	-	-	10.2218477
To AC	= 270	-	-	2.4313627

3. By Gunter.

Extend the compasses from $45^\{\backslash circ\}$ at the end of the tangents (the radius) to the tangent of $53^\{\backslash circ\}\; 7\text{'}\; 48\text{'}\text{'}$; then that extent will reach, on the line of numbers, from 162 to 216, for BC. Again, extend the compasses from $36^\{\backslash circ\}\; 52\text{'}$ to $90^\{\backslash circ\}$ on the fines; then that extent will reach, on the line of numbers, from 162 to 270 for AC.

Note. Another method, by making every side radius, is often added by the authors on trigonometry, which is thus: The given right-angled triangle being ABC, make first the hypotenuse AC radius, that is, with the extent of AC as a radius, and each of the centres A and C, describe arcs CD and AE (fig. 5.); then it is evident that each leg will represent the fine of its opposite angle, viz. the leg BC the fine of the arc CD or of the angle A, and the leg AB the fine of the arc AE or of the angle C. Again, making either leg radius, the other leg will represent the tangent of its opposite angle, and the hypotenuse the secant of the same angle; thus, with radius AB and centre A describing the

are BF, BC represents the tangent of that arc, or of the angle A, and the hypotenuse AC the secant of the same; or with the radius BC and centre C describing the arc BG, the other leg AB is the tangent of that arc BG or of the angle C, and the hypotenuse CA the secant of the same.

And then the general rule for all these cases is this, viz. that the sides bear to each other the same proportions as the parts or things which they represent. And this is called making every side radius.

SPHERICAL TRIGONOMETRY.

SPHERICAL TRIGONOMETRY is the art whereby, from three given parts of a spherical triangle, we discover the rest; and, like plane trigonometry, is either right-angled or oblique-angled. But before we give the analogies for the solution of the several cases in either, it will be proper to premise the following theorems:

THEOREM I. In all right-angled spherical triangles, the sign of the hypotenuse : radius :: fine of a leg : fine of its opposite angle. And the fine of a leg : radius :: tangent of the other leg : tangent of its opposite angle.

Demonstration. Let EDAFG (fig. 6.) represent the eighth part of a sphere, where the quadrantal planes EDFG, EDBC, are both perpendicular to the quadrantal plane ADFB; and the quadrantal plane ADGC is perpendicular to the plane EDFG; and the spherical triangle ABC is right-angled at B, where CA is the hypotenuse, and BA, BC, are the legs.

To the arches GF, CB, draw the tangents HF, OB, and the fines GM, CI, on the radii DF, DB; also draw BL the fine of the arch AB, and CK the fine of arc C; and then join IK and OL. Now HF, OB, GM, CI, are all perpendicular to the plane ADFB. And HD, GK, OL, lie all in the same plane ADGC. Also FD, IK, BL, lie all in the same plane ADGC. Therefore the right-angled triangles HFD, CIK, OLB, having the equal angles HDE, CKI, OLB, are similar. And CK : DG :: CI : GM; that is, as the fine of the hypotenuse : rad. :: fine of a leg : fine of its opposite angle. For GM is the fine of the arc GF, which measures the angle CAB. Also, LB : DF :: BO : FH; that is, as the fine of a leg : radius :: tangent of the other leg : tangent of its opposite angle. Q. E. D.

Hence it follows, that the fines of the angles of any oblique spherical triangle ACD (fig. 7.) are to one another, directly, as the fines of the opposite sides. Hence it also follows, that, in right-angled spherical triangles, having the same perpendicular, the fines of the bases will be to each other, inversely, as the tangents of the angles at the bases.

THEOREM II. In any right-angled spherical triangle ABC (fig. 8.) it will be, As radius is to the co-fine of one leg, so is the co-fine of the other leg to the co-fine of the hypotenuse.

Hence, if two right-angled spherical triangles ABC, CBD (fig. 7.) have the same perpendicular BC, the co-fines of their hypotenuses will be to each other, directly, as the co-fines of their bases.

THEOREM III. In any spherical triangle it will be, As radius is to the fine of either angle, so is the co-fine of the adjacent leg to the co-fine of the opposite angle.

Hence, in right-angled spherical triangles, having the same perpendicular, the co-fines of the angles at the base will be to each other, directly, as the fines of the vertical angles.

THEOREM IV. In any right-angled spherical triangle

Spherical. it will be, As radius is to the co-sine of the hypothenuse, so is the tangent of either angle to the co-tangent of the other angle.

As the sum of the sines of two unequal arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their difference: and as the sum of the co-sines is to their difference, so is the co-tangent of half the sum of the arches to the tangent of half the difference of the same arches.

THEOREM V. In any spherical triangle ABC (fig. 9 and 10.) it will be, As the co-tangent of half the sum of half their difference, so is the co-tangent of half the base to the tangent of the distance (DE) of the perpendicular from the middle of the base.

Since the last proportion, by permutation, becomes co-tang. $\backslash frac\{AC+BC\}\{2\}$ : co-tang. AE :: tang. $\backslash frac\{AC-BC\}\{2\}$ : tang.

DE, and as the tangents of any two arches are, inversely, as their co-tangents; it follows, therefore, that tang. AE : $\backslash frac\{AC+BC\}\{2\}$ :: tang. $\backslash frac\{AC-BC\}\{2\}$ : tang. DE; or, that the tangent of half the base is to the tangent of half the sum of the sides, as the tangent of half the difference of the sides to the tangent of the distance of the perpendicular from the middle of the base.

THEOREM VI. In any spherical triangle ABC (fig. 9.) it will be, As the co-tangent of half the sum of the angles at the base is to the tangent of half their difference, so is the tangent of half the vertical angle to the tangent of the angle which the perpendicular CD makes with the line CF bisecting the vertical angle.

The Solution of the CASES of right-angled Spherical Triangles, (fig. 8.).

Case	Given	Sought	Solution
1	The hyp. AC and one angle A	The opposite leg BC	As radius : sine hyp. AC :: sine A : sine BC (by the former part of theor. 1.)
2	The hyp. AC and one angle A	The adjacent leg AB	As radius : co-sine of A :: tang. AC : tang. AB (by the latter part of theor. 1.)
3	The hyp. AC and one angle A	The other angle C	As radius : co-sine of AC :: tang. A : co-tang. C (by theorem 4.)
4	The hyp. AC and one leg AB	The other leg BC	As co-sine AB : radius :: co-sine AC : co-sine BC (by theorem 2.)
5	The hyp. AC and one leg AB	The opposite angle C	As sine AC : radius :: sine AB : sine C (by the former part of theorem 1.)
6	The hyp. AC and one leg AB	The adjacent angle A	As tang. AC : tang. AB :: radius : co-sine A (by theorem 1.)
7	One leg AB and the adjacent angle A	The other leg BC	As radius : sine AB :: tangent A : tangent BC (by theorem 4.)
8	One leg AB and the adjacent angle A	The opposite angle C	As radius : sine A :: co-sine of AB : co-sine of C (by theorem 3.)
9	One leg AB and the adjacent angle A	The hyp. AC	As co-sine of A : radius :: tang. AB : tang. AC (by theorem 1.)
10	One leg BC and the opposite angle A	The other leg AB	As tang. A : tang. BC :: radius : sine AB (by theorem 4.)
11	One leg BC and the opposite angle A	The adjacent angle C	As co-sine BC : radius :: co-sine of A : sine C (by theorem 3.)
12	One leg BC and the opposite angle A	The hyp. AC	As sine A : sine BC :: radius : sine AC (by theorem 1.)
13	Both legs AB and BC	The hyp. AC	As radius : co-sine AB :: co-sine BC : co-sine AC (by theorem 2.)
14	Both legs AB and BC	An angle, suppose A	As sine AB : radius :: tang. BC : tang. A (by theorem 4.)
15	Both angles A and C	A leg, suppose AB	As sine A : co-sine C :: radius : co-sine AB (by theorem 3.)
16	Both angles A and C	The hyp. AC	As tang. A : co-tang. C :: radius : co-sine AC (by theorem 4.)

Note. The 10th, 11th, and 12th cases are ambiguous; since it cannot be determined by the data, whether A, B, C, and AC, be greater or less than 90 degrees each.

Fig. 1.

Fig. 3.

Fig. 2.

Fig. 4.

Fig. 5.

Spherical TRIGONOMETRY.

Fig. 6.

Fig. 7.

Fig. 9.

Fig. 10.

Fig. 8.

Fig. 11.

Fig. 12.

TURNING.

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

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The Solution of the Cases of oblique spherical Triangles, (fig. 9 and 10.)

Case	Given	Sought	Solution
1	Two sides AC, BC, and an angle A opposite to one of them.	The angle B opposite to the other	As $\backslash text\{fine\; BC\}\; :\; \backslash text\{fine\; A\}\; ::\; \backslash text\{fine\; AC\}\; :\; \backslash text\{fine\; B\}$ (by theorem 1.) Note, this case is ambiguous when BC is less than AC; since it cannot be determined from the data whether B be acute or obtuse.
2	Two sides AC, BC, and an angle A opposite to one of them	The included angle ACB	Upon AB produced (if need be) let fall the perpendicular CD; then (by theorem 4.) $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; AC\}\; ::\; \backslash text\{tang.\; A\}\; :\; \backslash text\{co-tang.\; ACD\}$; but (by theorem 1.) as $\backslash text\{tang.\; BC\}\; :\; \backslash text\{tang.\; AC\}\; ::\; \backslash text\{co-fine\; ACD\}\; :\; \backslash text\{co-fine\; BCD\}$. Whence $\backslash text\{ACB\}\; =\; \backslash text\{ACD\}\; =\; \backslash text\{BCD\}$ is known.
3	Two sides AC, BC, and an angle opposite to one of them	The other side AB	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; A\}\; ::\; \backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; AD\}$ (by theor. 1.) and (by theor. 2.) as $\backslash text\{co-fine\; AC\}\; :\; \backslash text\{co-fine\; BC\}\; ::\; \backslash text\{co-fine\; AD\}\; :\; \backslash text\{co-fine\; BD\}$. Note, this and the last case are both ambiguous when the first is so.
4	Two sides AC, AB, and the included angle A	The other side BC	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; A\}\; ::\; \backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; AB\}$ (by theor. 1.) whence AD is also known; then (by theor. 2.) as $\backslash text\{co-fine\; AD\}\; :\; \backslash text\{co-fine\; BD\}\; ::\; \backslash text\{co-fine\; AC\}\; :\; \backslash text\{co-fine\; BC\}$.
5	Two sides AC, AB, and the included angle A	Either of the other angles, suppose B	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; A\}\; ::\; \backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; AD\}$ (by theor. 1.) whence BD is known; then (by theor. 4.) as $\backslash text\{fine\; BD\}\; :\; \backslash text\{fine\; AD\}\; ::\; \backslash text\{tang.\; A\}\; :\; \backslash text\{tang.\; B\}$.
6	Two angles A, ACB, and the side AC betwixt them	The other angle B	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; AB\}\; ::\; \backslash text\{tang.\; A\}\; :\; \backslash text\{co-tang.\; ACD\}$ (by theorem 4.) whence BCD is also known; then (by theor. 3.) as $\backslash text\{fine\; ACD\}\; :\; \backslash text\{fine\; BCD\}\; ::\; \backslash text\{co-fine\; A\}\; :\; \backslash text\{co-fine\; B\}$.
7	Two angles A, ACB, and the side AC betwixt them	Either of the other sides, suppose BC	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; AC\}\; ::\; \backslash text\{tang.\; A\}\; :\; \backslash text\{co-tang.\; ACD\}$ (by theorem 4.) whence BCD is also known; then, as $\backslash text\{co-fine\; BCD\}\; :\; \backslash text\{co-fine\; ACD\}\; ::\; \backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; BC\}$ (by theor. 1.)
8	Two angles A, B, and a side AC opposite to one of them	The side BC opposite the other	As $\backslash text\{fine\; B\}\; :\; \backslash text\{fine\; AC\}\; ::\; \backslash text\{fine\; A\}\; :\; \backslash text\{fine\; BC\}$ (by theorem 1.)
9	Two angles A, B, and a side AC opposite to one of them	The side AB betwixt them	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; A\}\; ::\; \backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; AD\}$ (by theor. 1.) and as $\backslash text\{tang.\; B\}\; :\; \backslash text\{tang.\; A\}\; ::\; \backslash text\{fine\; AD\}\; :\; \backslash text\{fine\; BD\}$ (by theorem 4.) whence AB is also known.
10	Two angles A, B, and a side AC opposite to one of them	The other angle ACB	As $\backslash text\{rad.\}\; :\; \backslash text\{co-fine\; AC\}\; ::\; \backslash text\{tang.\; A\}\; :\; \backslash text\{co-tang.\; ACD\}$ (by theorem 4.) and as $\backslash text\{co-fine\; A\}\; :\; \backslash text\{co-fine\; B\}\; ::\; \backslash text\{fine\; ACD\}\; :\; \backslash text\{fine\; BCD\}$ (by theor. 3.) whence ACB is also known.
11	All the three sides AB, AC, and BC	An angle, suppose A	As $\backslash text\{tang.\; \}\; \backslash frac\{1\}\{2\}AB\; :\; \backslash text\{tang.\; \}\; \backslash frac\{AC+BC\}\{2\}\; ::\; \backslash text\{tang.\; \}\; \backslash frac\{AC-BC\}\{2\}\; :\; \backslash text\{tang.\; DE\}$, the distance of the perpendicular from the middle of the base (by theorem 6.) whence AD is known; then, as $\backslash text\{tang.\; AC\}\; :\; \backslash text\{tang.\; AD\}\; ::\; \backslash text\{rad.\}\; :\; \backslash text\{co-fine\; A\}$ (by theor. 1.)
12	All the three angles A, B, and ACB	A side, suppose AC	As $\backslash text\{co-tang.\; \}\; \backslash frac\{ABC+A\}\{2\}\; :\; \backslash text\{tang.\; \}\; \backslash frac\{ABC-A\}\{2\}\; ::\; \backslash text\{tang.\; \}\; \backslash frac\{ACB\}\{2\}\; :\; \backslash text\{tan.\; of\; the\; angle\; included\; by\; the\; perpendicular\; and\; a\; line\; bisecting\; the\; vertical\; angles\}$; whence ACD is also known; then (by theorem 5.) $\backslash text\{tang.\; A\}\; :\; \backslash text\{co-tang.\; ACD\}\; ::\; \backslash text\{rad.\}\; :\; \backslash text\{co-fine\; AC\}$.

The following propositions and remarks, concerning spherical triangles (selected and communicated to Dr Hutton by the reverend Nevil Maskelyne, D. D. Astronomer Royal, F. R. S.), will also render the calculation of them peripicuous, and free from ambiguity.

1. A spherical triangle is equilateral, isosceles, or scalene, according as it has its three angles all equal, or two of them equal, or all three unequal; and vice versa.

2. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle.

3. Any two sides taken together are greater than the third.

4. If the three angles are all acute, or all right, or all obtuse; the three sides will be, accordingly, all less than $90^\backslash circ$, or equal to $90^\backslash circ$, or greater than $90^\backslash circ$; and vice versa.

5. If from the three angles A, B, C, of a triangle ABC, as poles, there be described, upon the surface of the sphere, three arcs of a great circle DE, DF, FE, forming by their intersections a new spherical triangle DEF; each side of the new triangle will be the supplement of the angle at its pole; and each angle of the same triangle will be the supplement of the side opposite to it in the triangle ABC.

6. In any triangle ABC, or A'C, right-angled in A, if, Fig. 12. The angles at the hypothenuse are always of the same kind

Spherical as their opposite sides; 2dly, The hypothenuse is less or greater than a quadrant, according as the sides including the right angle are of the same or different kinds; that is to say, according as these same sides are either both acute or both obtuse, or as one is acute and the other obtuse. And vice versa,

Spherical 1st, The sides including the right angle are always of the same kind as their opposite angles: 2dly, The sides including the right angle will be of the same or different kinds, according as the hypothenuse is less or more than 90°; but one at least of them will be of 90°, if the hypothenuse is so.

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Tribulate
Tringa TRIHILATÆ, from tres "three," and bilum "an external mark on the feed;" the name of the 23d class in Linnaeus's Fragments of a Natural Method; consisting of plants with three seeds, which are marked with an external cicatrix or scar, where they are fastened within the fruit. See BOTANY, Sect. 6.

TRIM, implies in general the state or disposition by which a ship is best calculated for the several purposes of navigation.

Thus the trim of the hold denotes the most convenient and proper arrangement of the various materials contained therein relatively to the ship's motion or stability at sea. The trim of the masts and sails is also their most apposite situation with regard to the construction of the ship and the effort of the wind upon her sails. See SEAMANSHIP.

TRINGA, SANDPiper; a genus of birds belonging to the order of grallæ. The bill is somewhat tapering, and of the length of the head; the nostrils are small; the toes are four in number and divided, the hind toe being frequently raised from the ground. According to Dr Latham there are 45 species, of which 18 are British. We shall describe some of the most remarkable.

1. Vanellus, lapwing, or tewit, is distinguished by having the bill, crown of the head, crest, and throat, of a black colour; there is also a black line under each eye; the back is of a purplish green; the wings and tail are black and white, and the legs red: the weight is 8 ounces and the length 13 inches. It lays four eggs, making a flight nest with a few bends. The eggs have an olive cast, and are spotted with black. The young, as soon as hatched, run like chickens: the parents show remarkable solicitude for them, flying with great anxiety and clamour near them, striking at either men or dogs that approach, and often fluttering along the ground like a wounded bird, to a considerable distance from their nest, to delude their pursuers; and to aid the deceit, they become more clamorous when most remote from it: the eggs are held in great esteem for their delicacy, and are sold by the London poulterers for three shillings the dozen. In winter, lapwings join in vast flocks; but at that season are very wild: their flesh is very good, their food being insects and worms. During October and November, they are taken in the fens in nets, in the same manner that ruffs are; but are not preserved for fattening, being killed as soon as caught.

2. Pagax. The male of this species is called ruff, and the female reeve. The name ruff is given to the males because they are furnished with very long feathers, standing out in a remarkable manner, not unlike the ruff worn by our ancestors. The ruff is of as many different colours as there are males; but in general it is barred with black; the weight is six or seven ounces; the length, one foot. The female, or reeve, has no ruff; the common colour is brown; the feathers are edged with a very pale colour; the breast and belly white. Its weight is about four ounces.

These birds appear in the fens in the earliest spring, and disappear about Michaelmas. The reeves lay four eggs in

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a tuft of grass, the first week in May, and sit about a month. The eggs are white, marked with large rusty spots. Fowlers avoid in general the taking of the females; not only because they are smaller than the males, but that they may be left to breed.

Soon after their arrival, the males begin to hill, that is, to collect on some dry bank near a splash of water, in expectation of the females, who resort to them. Each male keeps possession of a small piece of ground, which it runs round till the grass is worn quite away, and nothing but a naked circle is left. When a female lights, the ruffs immediately fall to fighting. It is a vulgar error, that ruffs must be fed in the dark lest they should destroy each other by fighting on admission of light. The truth is, every bird takes its stand in the room as it would in the open fen. If another invades its circle, an attack is made, and a battle ensues. They make use of the same action in fighting as a cock, place their bills to the ground and spread their ruffs. Mr Pennant says, he has set a whole room-full a-fighting, by making them move their stations; and after quitting the place, by peeping through a crevice, seen them resume their circles and grow pacific.

When a fowler discovers one of those hills, he places his net over night, which is of the same kind as those that are called clap or day nets; only it is generally single, and is about 14 yards long and four broad. The fowler resorts to his stand at day-break, at the distance of one, two, three, or four hundred yards from the nets, according to the time of the season; for the later it is, the shyer the birds grow. He then makes his first pull, taking such birds as he finds within reach: after that he places his stuffed birds or stales to entice those that are continually traversing the fen. When the stales are set, seldom more than two or three are taken at a time. A fowler will take 40 or 50 dozen in a season. These birds are found in Lincolnshire, the Isle of Ely, and in the East Riding of York. They visit a place called Martin-Mere in Lancashire the latter end of March or beginning of April; but do not continue there above three weeks; where they are taken in nets, and fattened for the table with bread and milk, hempseed, and sometimes boiled wheat; but if expedition is required, sugar is added, which will make them in a fortnight's time a lump of fat: they then sell for two shillings or half a crown a-piece. They are dressed like the woodcock, with their intestines; and when killed at the critical time, say the Epicures, are the most delicious of all morsels.

3. Canutus, or knot, has the forehead, chin, and lower part of the neck, brown, inclining to ash-colour; the back and scapulars deep brown, edged with ash colour; the coverts of the wings white, the edges of the lower order deeply so, forming a white bar; the breast, sides, and belly white, the two first streaked with brown; the coverts of the tail marked with white and dusky spots alternately; the tail ash coloured, the outmost feather on each side white; the legs of a bluish grey; and the toes, as a special mark, divided to the very bottom; the weight four ounces and a half.—