If Ec be supposed to represent a part of the ecliptic, then BE and BC will be quadrants (H.133.) and

CEB and ECB will be right angles (I. 134.)

In the right-angled triangle CEA, making EA the middle part,

Crad

COS C

sine EA

sine b

xb, and because the altitude of the nonagesimal degree of the ecliptic, is an arc of a great circle comprehended between the zenith of any place of the pole of the ecliptic (R. 265.), we obtain the following proportion.

(W) Sine of the zenith distance is to cosine of the altitude of the nonagesimal degree; as the parallax in altitude, is to the parallax in latitude, viz. sine b: sine EA=cos c::b. a.

Again, in the right-angled triangle CEA, making the ECA the middle part, we have rad x cos ECA=tang ECX cot b; but

rad2 b

(O. 104.) hence

COS ECA

sine c
rad

and by substitution B =

tang EC
tang b

tang EC tang b

xb (because the planet is sup

posed to be in or very near to the ecliptic, sine a = rad) hence,

(X) Tangent of the planet's zenith distance, is to the tangent of its longitude from the nonagesimai degree; as the parallax in altitude, is to the parallax in longitude, viz.

(Y) Given the altitude of the nonagesimal degree of the ecliptic; the longitude of a planet from the nonagesimal degree, and its horizontal parallax, to find its parallax in latitude and longitude.

It is shewn in the preceding proposition that a =

and B=

sineb

tang Ec tang b rallax of any planet, its parallax in altitude ¿= XH nearly.

xb. Now, if H represent the horizontal pa

sineb
rad

(T. 96.) By substituting this value of 6 in the above equations

COS C

× H, and B

tang EC x sine b

tang bx rad

X H.

rad In the right-angled triangle CEA, making b the middle part, rad x cos b COS EA X COS EC; but cos b=

rad x sine b

and

tang b

COS EA X sine EC

rad2

× H, but Ec is the measure of the в (L.227.)

and cos EA sine o, therefore B

sine C X sine B

× H; hence are

rad2

derived the following general rules.

1. Radius, is to the cosine of the altitude of the nonagesimal degree of the ecliptic; as the horizontal 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.

(Z) 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 declin

* Simpson's Fluxions, vol. ii. page 286.

22

ation 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 B to represent 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

c

B

m c

Now we have shewn (P. 354.) that ab:: sine a cot c; (sine a x cot c)-(cos a x cos B)

But cot c

sine B

(O. 189.)

therefore by substitution, and dividing by sine of a,

:-),

sine a

sine B

sine a

rad

cot B

rad

cot c

cot a

sine B 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.*

PROPOSITION XIII.

(A) 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 в will be variable. It is shewn (P. 354.) that B:b::rad: 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 ▲ (if

* Astronomie Nautique, par M. de Maupertuis, page 34. metry, second edition, page 144, &c.

Vince's Trige

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.*

PROPOSITION XIV.

(B) The error in the altitude of any tower, or other object, is to the error committed in taking the angle of elevation; as double the height of the observed object, is to the sine of double the angle of elevation.

This proposition is designed to illustrate

the scholium (S. 355.). From the first set of

equations (C. 347.) we have A=

sine c

sine b

that is, a A: sine b: sine c, and considering the triangle as rectilinear (S. 355.)

But, sine 2A 2 sine A X cos A, rad 1 (O. 108.)

Again, sine aa::rad=1:b (Z. 34.)
And sine a

2a::1: 2b 2b. sine A 2a, by substitu

that is, a: A:: 2a: sine 2a which was to be sine 2 A

a

2 a

tion = Á

shewn.

COROLLARY.

Because a =

ix 2α
sine 2A

it is obvious that (a)

the error in altitude will be a minimum, when sine 2A is a maximum, that is sine of 90°. Hence if (A) the error in the observed altitude be 1', the sine or arc of which is 0002909, and the observed angle be 45°, we have

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

CHAP. XIV.

MISCELLANEOUS PROPOSITIONS, &c.

(C) 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.

A 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 x 60 to 100 x 100, or of 27 to 50; and a French second is less than an English second in the ratio of 90 x 60 x 60 to 100 x 100 x 100 or of 81 to 250.

Hence, if nany number of degrees, to turn English de

grees into French, we have 9: 10::n:

10n

9

9n

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

10

n

n+,and to turn

9

PROPOSITION I.

(D) 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

Preface

* Eléments de Géométrie, par A. M. Legendre, 6th ed. page 328. 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 279.07.85′′, or 27°.0735; 45°.18.4′′ — 45°. 18′04′′ — 45°.1804, &c.