NOTES.

NOTE (A). page 53.

We are to shew here that in the formula of De Moivre, viz.

the first member has m values as well as the second. This fact we shall easily establish, by means of the property adverted to in the text, viz. that to any given values of the lines sin. A, cos. A, there correspond innumerable different arcs, viz. every arc in the infinite series,

A, 2π + A, 4π + А, 6π + A, &c.

so that the first member of the above formula involves in it the following values, viz.

These values will continue different till we arrive at such a value, N, for

one of the numeral coefficients, 1, 2, 4, 6, &c. as will render

multiple of 2π, when the first of the foregoing values will obviously

recur, so that by continuing the series we shall merely obtain a repetition

n

of the former values. Now Nπ cannot become a multiple of 2π till N

m

become equal to 2m; hence we shall have expressed all the different values involved in the first member of De Moivre's formula when we have continued the above series of values as far as that in which the numeral coefficient is 2m 2; that is, when we have written m values. Hence each member of the formula involves m different values.

NOTE (B), page 108.

Professor Vince, at page 43 of his Trigonometry, has the following

note.

"Difficulties have frequently arisen in consequence of its being supposed that an arc of 90° has a tangent and secant, each infinite. For instance, in a right-angled spherical triangle, radius: cosine of the angle at the base tangent of the hypothenuse: tangent of the base; now when the base = 90°, the hypothenuse = 90°; and, therefore, these arcs being equal, if they have any tangents, of whatever value they may be, they must be equal; and, therefore, radius cosine of the angle at the base, whatever that angle may be. This false conclusion arises from the supposition that an arc increases till it becomes 90°; the tangent and secant increase without limit; and at 90° the arc ceases to have either a tangent or secant, by their definition. As the arc, by increasing, passes through 90°, the tangent and secant increase without limit, cease to exist at 90°, and then begin again at a quantity indefinitely great. And thus in other cases where the tangent or secant of an arc enter into the computation, when the arc becomes 90°, we can draw no conclusion on which we can depend."

The foregoing reasoning is very much calculated to mislead the young student, although it does in reality tend to overturn the author's own hypothesis, and to show that the tangent of 90° must necessarily be infinite.

Taking the example chosen above, by Mr. Vince, we have for the true solution

which must necessarily involve the absurdity noticed above, except tan. 90° be either 0 or ; but when the proper value is put for tan. 90°, then we have

and as

0

admits not only of the particular value 1, fixed upon by Mr. Vince, but of an indefinite number of values, so does cos. Zat base. Upon the same grounds that Mr. Vince has rejected the tangent of 90°, he should have rejected the cosine of 90°, which, however, he admits to be 0.

For

sin. at base = rad.

cos.at vertex
cos. base

but, when both base and hypotenuse are 90°, the angle at the vertex is 90°, and we ought therefore to have, according to Mr. Vince, sin. at baserad, which is, indeed, one solution, but by no means the only one, because the values of are innumerable.

NOTE (C), page 130.

It was shewn, at page 128, that if a ship in latitude x, vary her latitude by a very small portion ▲, and that she continue her course till her departure equals the difference of longitude due to the difference of latitude Ar, then the enlarged difference of latitude (Ay), due to this departure,

will be

Ay sec. x At ...

ду
AT

sec. 1.

This expression, it must be remembered, is nearer the truth the smaller we suppose ▲r to be, and is, therefore, accurately true only when Ax=0;

in other words, sec. x is the value to which the ratio

proaches as we continually diminish ▲r, (and in consequence Ay,) and which value it actually becomes only when the terms of the ratio vanish,

.. y=log. tan. (45° + x), see Int. Calculus, p. 69; the logarithm here used is the Naperian. To change it into a common logarithm we must multiply by the modulus 2.302585, &c.; it must be observed, however, that it is the logarithm of the natural tangent, which is here expressed, and not the tabular logarithmic tangent; it is, therefore, equal to the tabular logarithmic tangent minus 10. Hence, employing the table of logarithmic tangents, we may compute y from the formula y=2·302585 {log. tan. (45° + 1⁄2 x) — 10} × Rad.

and thus, as stated in the text, the meridional parts, y, corresponding to any given latitude x, may be expeditiously computed, independently of any previous computations.

The tables of meridional parts are usually expressed in nautical miles, and we shall have the number of miles in y, if, instead of multiplying by the radius of the earth, we multiply by the number of miles or minutes in it. Now in every circle the radius is equal to 3437-74679 minutes of that circle, because

3.14159, &c. 180°:: 1: 3437-74679 minutes; hence, for the number of miles in y the expression is

7915 7044679 {log. tan. (45° + x) — 10} ;

or, since tan. 45° + } x = cot. 45° — 1⁄2 x, and, since, moreover,

this expression may be written thus,

7915-7044679 {10 — log. tan. (45° — } x)},

which gives the rule in the text.

We had intended to have introduced here some other particulars relating to Mr. Wright's projection of the meridian line, but we are precluded from doing so, as this treatise has already exceeded the limits assigned to it. We must, therefore, content ourselves with referring the student to Robertson's Navigation, vol. 11. p. 135–146.

NOTE (D), page 220.

The following pretty theorems I have received from Mr. Lowry, since the first chapter on Spherical Geometry was in forms.

"Let ABC be a spherical triangle,* D the middle of one of the sides, AC; and let AB d. Then

Cor. 1. When the triangle is inscribed in a semi-circle, the diameter of which is b,

cos.a+cos. c = 2 cos.2 b, or cos. a+ cos. c = 1 + cos. b.

Cor. 2. And when a = c, we have cos. a=cos.2.

2

Cor. 3. Hence, in a spherical square,† the cosine of the sides is equal to the square of the cosine of half the diagonal.

* The figure may be easily sketched by the student.

✦ A spherical four sided figure, whose sides are all equal, and whose angles are also all equal.