1. Radius, is to the cosine of the altitude of the nonagesimal degree of the ecliptic; as the horizontul parallax, is to the parallax in latitude.

2. The square of the radius, is to the rectangle of the sines of the altitude of the nonagesimal degree and the planet's longitude from thence; as the horizontal parallax, is to the parallax in longitude*.

PROPOSITION XII.

(0) To determine the correction for finding the time of apparent noon, from equal altitudes of the sun.

It is obvious that if the sun's declination were invariable, half the interval of time between equal altitudes would shew the instant of noon; but by the variation in the sun's declination he will have the same altitude at different distances from the meridian; this variation will, in general, be very small and can only affect the polar distance.

If therefore, we suppose в to repre'sent the pole of the equinoctial, A the zenith, and c the place of the sun; AB and AC will be constant quantities, and BC variable.

A

b

C

B

Now we have shewn (F. 318.) that a (sine ax cot c)-(cos a x cos B) But cot c=

sine B

therefore by substitution, and dividing by

B: sine a cot c;

(Q. 219.),

sine of a,

(

cot c

sine B

(O. 98.) =

cot a

tang B). The half of this expression, re

duced into time, will be the correction required, where cot c= tangent of the latitude, and cot a=tangent of the declination †.

Simpson's Fluxions, vol. II. page 286.

+ Astronomie Nautique, par M. de Maupertuis, page 34.

nometry, second edition, page 144, &c.

Vince's Trigo

PROPOSITION

PROPOSITION XIII.

(P) The error in taking the altitude of a star being given, to find the corresponding error in the hour angle.

As in the preceding proposition, let в represent the pole of the equinoctial, A the zenith, and c the observed place of the star. Then c will be the co-latitude, a the star's co-declination, and b its co-altitude; the sides c and a will be constant quantities, and the hour angle B will be variable. It is shewn (F. 318.) that B: b:: rad2: sine A × sine c, hence

Since the variation of the angle B is the measure of the error in time, and that c, in the same latitude, and A (if the same error. prevails in different observations) are constant quantities; the error in time will not be altered whatever the altitude of the star may be, and this error will be the least if the altitude of the star be taken when it is on the prime vertical *.

SCHOLIUM.

(Q) In all the preceding propositions if the sides of the triangle be diminished without limit, the triangle may be considered as rectilinear, and instead of the sines and tangents of the sides, we may substitute the sides themselves (B. 223.). Hence the variations of plane triangles are readily deduced from those of spherical triangles, in every case where the fluxions are proportional to the sines or tangents of the sides. Thus, by the 4th proportion (F. 318.) we have shewn that a : ċ:: rad x sine a : cos B X sine c, that is, (supposing the triangle straight lined) ▲ : ċ :: rad x a:

COS B X C.

Again, by the 8th proportion (H. 319.) b:c:: tang b: tang c, that is, when the triangle is straight lined, b: ċ :: b: c, and in the same manner the rest may be deduced.

The variations of rectilinear triangles may be deduced from the triangles themselves, without reference to spherical triangles, in a manner exactly similar to those deduced from the spherical triangles. Vide Traité de Trigonométrie, par M. Cagnoli, chapitre X.

* See Dr. Mackay's Theory and Practice of finding the longitude at sea or land, vol. I, page 298, third edition. Tt 2

CHAP.

CHAP. X.

MISCELLANEOUS PROPOSITIONS, &c.

(R) 1. Of the French Division of the Circle.

The modern French writers on Trigonometry divide the circumference of the circle into 400* equal parts or degrees, each degree into 100 equal parts or minutes, each minute into 100 equal parts or seconds, &c. which degrees, minutes, &c. they write in the usual manner, thus, 126°.80'.64", &c.

Á French degree is therefore less than an English degree, in the ratio of 90 to 100, or of 9 to 10; a French minute is less than an English minute in the ratio of 90 × 60 to 100 × 100, or of 27 to 50; and a French second is less than an English second in the ratio of 90 × 60 × 60 to 100 x 100 x 100 or of 81 to 250. Hence, if nany number of degrees, to turn English de

10n

grees into French, we have 9: 10 : : ~ : =n+ , and to turn

9

9'

n

French degrees into English, 10:9::n:

PROPOSITION I.

(S) To turn French degrees, minutes, &c. into English.

RULE. Consider the degrees as a whole number, after which place the minutes and seconds † as decimals; of this mixed decimal deducted from itself will give the English degrees corresponding to the French.

EXAMPLE I. The latitude of Paris is 54°.26'.36" in the

* Eléments de Géométrie, par A. M. Legendre, 6th ed..page 328. Preface to Borda's Trigonometrical Tables (Paris, An. IX.), page 18, et seq. + The minutes and seconds if under 10, must have a cipher prefixed, thus 27.7.35", must be written 27°.07′.35′′, or 27°.0735; 45°.18′.4′′—45°.18′.04′′= 450.1804, &c.

French

French division of the circle, what is the corresponding latitude in the English division?

54°.26.36"54°.2636

French degrees.

I = 542636

48 83724 English degrees.

60

50 23440

60

14 06400

Answer. 480.50'.14" English.

2. What number of degrees, &c. in the English division of the circle will correspond to 74.4.8" in the French division. 74°.4.8" 74°.04.08"=74°0408

7.40408

66.63672

60

38.20320

60..

12.19200

Answer. 66.38'.12" English.

PROPOSITION II.

(T) To turn English degrees, minutes, &c. into French.

RULE. Reduce the minutes and seconds, &c. to the decimal of a degree, and annex it to the given number of degrees; this mixed decimal increased by of itself will give the French degrees corresponding to the English.

EXAMPLE. The latitude of Greenwich Observatory is 51°.28'.40" N., according to the English division of the circle, what is the corresponding latitude by the French division? 51° 28'.40 = 51°.477777, &c.

Answer. 570.19.75".3 North, by the French division of

the circle.

On

On the 22d March, 1813, the moon's distance from the sun, at midnight, will be 114°.13.21" by the Nautical Almanac, what will be the distance according to the French division of the circle?

(U) A table of arcs differing by 10 degrees according to the French division of the circle, or by 9 degrees by the English division, with the corresponding natural and logarithmical

The latitudes of the Observatories of Paris and Pekin are 54°.26'.36" N. and 44°.33'.73" N., and their difference of longitude 126.80.56", according to the French division of the circle, what is their distance*?

(W) SOLUTION BY THE FRENCH DIVISION OF THE CIRCLET.

Here we have two sides a and b of a spherical triangle given, and the included angle c, to find the third side c.

b=(100°-44°.33.73")=55°.66'.27", and c=186°.80'.56"log cos c=log cos 126°.80.56" log cos 200°-126°.80′.56′′= log cos 73°.19.44".