NOTES.

NOTE (a), page 1.

THOUGH the centesimal division of the quadrant was immediately succeeded by corresponding trigonometrical tables of the most extensive scale, yet it was not attempted to be followed by any other nation: and even in France, it required all the authority of the government, and all the influence of Laplace, and other celebrated mathematicians by whom it was introduced, for many years to effect even its partial adoption. This might, however, have been expected from a project, notwithstanding its many and obvious advantages, that rendered useless all the trigonometrical and navigation tables, which, from having passed through so many hands, might be considered correct; that lessened, in a considerable degree, the value of the best instruments in observatories; and which at once sacrificed all the conveniences attending the continual trisection through the quadrant, so important to artists in the division of circular instruments. Indeed it would be difficult to imagine a more violent and extensive change than its universal adoption would have effected in all astronomical, mathematical, and geographical works. Its fate, however, may now be considered decided; the French mathematicians are gradually returning to the sexagesimal division, their power, gigantic as it was, having been found to be unequal to the task of destroying the prejudices, and altering the habits of a whole community.

To compare grades and degrees, we must observe that the circumference of the circle is divided into 400 and 360 equal parts respectively: hence a grade is to a degree as 9 is to 10. And the number of grades is to the number of degrees as 10 is to 9;

number of grades.

and, number of degrees=2 or (1-1)

Example I. Convert 638 66197 into degrees.

63.66197

one tenth 6·366197

57.295773

57° 17′ 44′′· 7828

Example II. Reduce 57° 17′ 44′′ · 7828 into grades.

57° 295773

one ninth 6 366197

63% 661970

NOTE (4), p. 73.

It seems not very easy to assign the reason why Napier, in framing his celebrated rules called the "Five Circular Parts," employed the complements of the hypothenuse and the complements of the angles, instead of the complements of the sides. Had he adopted the latter, and altered his rules to the following:-product of radius and cosine of the middle part

product of cotangents of the conjoined parts, and product of sines of the disjoined parts; they would have been more convenient in practice, since there would be fewer changes to make relating to the complements.

The science of Trigonometry is greatly indebted to this great man. In addition to his discovery of logarithms, by the aid of which the calculations which occupied months could be performed in as many days; and the final simplification of the decimal notation, he discovered the four equations in (90.) and (94.) which are commonly known by the name of Napier's Analogies. His discoveries also seem to be attended with the singular felicity of admitting of no improvement.

NOTE (c), pp. 33. 87.

It will be observed, in the solutions of plane and spherical triangles, that we have given severa formulæ for the same case. The reason of this is, that an angle cannot be accurately determined from its cosine when very small, nor from its sine when near 90°; but from its tangent it can generally be found with accuracy. Of the expressions, therefore, in (50.)

and (51.), the first must not be employed when

small; nor the second when

2

C 2

or C is very

is near 90°; but that in (52.)

(except in extreme cases to be mentioned) may be always employed. The same observations apply when the expressions in (89.) are used.

In (47.) we have the expression cos A

curate if A is small, but we have,

which is inac

c