TRACT 19.

TRIGONOMETRICAL TABLES, &.C.

289

then the series of the chords and supplemental chords of themultiple arcs will be thus ; where the values are alternately

Arcs

Chords and Chords of Sup.

la

Chord = c

2 a

Sup. ch. = — c 1 + 2

3 a

Chord = — c 3 + 3c

4 a

Sup. ch. = + c 4 — 4c 2 + 2

5a

Chord = + c s - 5c 3 + 5c

6a

Sup. ch. = - c 6 + 6c 4 - 9c 2 + 2

la

Chord = — c 7 7c s — 14c 3 '+ 1c

&c.

&c.

chords, and chords of the supplements of the arcs on thesame line, and the law of the powers and coefficients as be-fore, but every alternate couplet of lines having their signschanged.

Another curious theorem is added to the above, for findingthe sum of all these chords drawn in a semicircle, from oneend of the diameter to every point in the circumference,those points dividing the circumference into any number ofequal parts ; namely, as the least chord is to the diameter, sois the sum of the said least chord and diameter and greatestchord, to double the sum of all the chords, including thediameter as one of them.

As the above theorems are chiefly adapted for the chordsof

multiple angles, a few problems and remarks are thenadded (whether by Vieta or Anderson does not clearly ap-pear, but I think by the latter) concerning the application ofthem, to the section of angles into submultiples, and thenceto the computation of the chords or sines, or a canon of tri-angles. The general precept for the angular sections is this:select one of the above equations adapted to the proper num-ber of the section, in which will be concerned the powers ofthe unknown or required quantity, as high as the index ofthe section; and from this equation find that quantity by theknown methods for the resolution of equations. Examples

vol. i. u