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= 1, According to Obs. 2, ART. 15.

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The results obtained in this Observation may be collected together for reference in

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This Table is to be made use of in the same way as TABLE II. for determining the magnitude of angles, and also for constructing another, in which the sines, cosines, &c. of several angles should be registered, in order to determine the magnitude of such angles when we happen to know that of their sine, cosine, &c.

We have thus deduced the elements of goniometry on the numerical as well as the geometrical system, in a separate and independent manner. This has undoubtedly lengthened our Treatise; but we hope it will prevent that confusion which is apt to arise in the mind of a reader upon finding the two systems used indiscriminately, as they are in most works on the science.

Obs. 1. As we have only to blot out the R in TABLES I. and II. in order to get the formulæ of TABLES III. and IV. so, reversely, if we would obtain a geometrical formula corresponding to any numerical formula in this science, we have only to introduce the linear radius, R, into the latter, according to TABLES I. and II. Thus, if we have the numeral formulæ,

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In order to obtain the corresponding geometrical formulæ, we have only to substitute the geometrical radius, R, for the numeral radius 1, as follows:

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The geometrical formulæ are immediately discoverable from the numerical, without referring to the Tables, if we are always careful to introduce the radius R, and its powers, so that the resulting formula may have all its terms of the same species of quantity, i. e. all lines, or all surfaces, &c. Thus having the formula,

sec2 A tan2 A + 1,

we immediately obtain the corresponding geometric formula (without consulting TABLE I.), by introducing — not R instead of 1; for then we should have

sec2 A tan2 A + R;

that is, a square equal to another square + a right line, which would be absurd,-but R2, as in this case the resulting formula would be

sec2 A = tan2 A + R2;

that is, a square equal to the sum of two squares, which is a rational equation.

Obs. 2. The sines, cosines, &c., heretofore spoken of, are called natural sines, cosines, &c. But these are now seldom registered in mathematical tables; instead of them we register their logarithms, which is found to be more convenient in practice. The logarithms of the sines, cosines, &c. of angles, are called the logarithmic sines, cosines, &c., of these angles; and being put opposite their corresponding

E

angles in the tables, are used instead of the natural sines, &c., with great advantage, as will be seen hereafter.

We may calculate the logarithms of the natural sines, &c., from these sines themselves; but for this purpose it is found most eligible to take the numeral radius = 10,000,000,000, or 1010, instead of 1, and to consider the natural sines, &c., as made up of submultiple parts of this numeral radius, in the same manner as we before considered them as made up of submultiple parts of 1*. In fact 1010, or any other number, may be looked upon as the unit, or whole, represented by 1, and therefore we may introduce 1010, or the said number, as radius, into the formulæ of ARTS. 16 and 17, merely recollecting to square the new radius wherever 1 has been squared, though its square be not expressed, inasmuch as 12 = 1.

N.B. When there is no especial purpose, such as the above, to be served, 1 is always chosen as numeral radius ; because its powers being all 1 disappear in computa

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tion, which is thereby much abbreviated.

LESSON V.

TRIGONOMETRY.

ART. 18. In a triangle the sides are to each other as the

sines of their opposite angles.

Let ABC be any triangle, of which the three sides are represented severally by a, b, c, and the three corresponding opposite angles by A, B, C.

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Then

sin a sin c,
sin A: sin B, Á

sin B sin c.

B

a

* If 1 were taken as radius, the sines and cosines of almost all angles being less than the radius, would be fractions, and therefore would have their logarithms subtractive numbers. (ART. 62, LOG.) This, amongst others, is the reason we choose a number so much greater than 1 for radius as will render the sines and cosines of all but indefinitely small angles greater than unity, so that their logarithms will be additive numbers.

DEM. Represent the perpendicular from the vertex of any angle в on the opposite side by y.

Obs. 2, LESSON IV, we shall have

According to

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In like manner it is shown that sin A: sin в :: a ; b, and sin B sin cbc, by drawing perpendiculars from the vertices of the other angles c and A to the other sides c and a. This, &c.

ART. 19. In a triangle the tangent of half the sum of any two angles is to the tangent of half their difference, as the sum of the opposite sides is to their difference.

Let ABC be a triangle, whose sides and angles are represented, as above, by a, b, c, and A, B, C.

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DEM. Produce BA, which is not the greater of the two sides under consideration, until it be equal to the other BC; join CD,

A

F

D

a

and let fall on it the perpendicular BE, which divides it equally at E, by ART. 139, GEOM.. The angle BCD = BDC (ART. 4, GEOM.); therefore BDC BCA + ACD. But by ART. 35, GEOM., Bac = BDC + ACD, .'. BAC = BCA + 2ACD,. .. ACD =

BAC-BCA

2

A-C

=

2

Likewise, as BAC =

BDC + ACD, adding the angle BCA to both, we get BAC + BCA BDC + ACD + BCA = 2BDC = 2BCD .'. BCD = BAC + BCA

2

A+ C
2

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2BA 2AF BD + BA BC + BA,,'. BF =

_atc

a + c

= 2

BC + BA 2

Finally: Joining FE, by ART. 161, GEOM., FE is parallel to AC. Consequently BE bE BF AF, by ART. 110, GEOM.; that is, representing the perpendiculars BE and be respectively by y and y', as also the line Ec by x, we have yy BF AF,

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ART. 20. In a triangle the cosine of any angle multiplied by twice the product of the sides which contain it, is equal to the sum of the squares of those sides diminished by the square of the third side.

Let ABC be a triangle, whose sides and angles are represented as above by a, b, c, and A, B, C. Then, cos A × 2bc = b2 + c2

a2.

B

a

E

b D

C

DEM. Represent the perpendicular BD by y, and represent the intercept AD by x.

The intercept DC

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