Theory of the Parallaxes of Altitude for the Sphere, by Mr. F. Mallet, Professor and Astronomer at Upsal; Translated from the French by M. Maty, M. D. R. S. Sec.

XXIX. Theory of the Parallaxes of Altitude for the Sphere, by Mr. F. Mallet, Professor and Astronomer at Upsal; Translated from the French by M. Maty, M.D.R.S. Sec. Upsal, October 25, 1766. Read Nov. 20, § 1. LET P be = the moon's horizontal parallax, or 1 to sin. P, as the moon's distance to the radius of the terrestrial sphere, on which the spectator is supposed to be placed. Let A be the distance of the moon from the zenith, and \( p \) the parallax of altitude for the same distance. The astronomers usually compute the value of \( p \) in the following manner: let sin. \( p = \sin. P \). Sin. A, and \( p' \) being found by the tables of logarithmic sines, sin. \( p'' = \sin. P \). Sin. A + \( p' \) is found in like manner, \( p'' \) being assumed for the true parallax, which is not accurate. § 2. In order to shew this, I have given another method of computing the parallax of altitude, as exactly as may be, by means of the common tables, in the following manner. Since sin. \( p = \sin. P \) sin. A + \( p \), we have sin. \( p = \sin. P \) sin. A cos. \( p + \sin. P \) cof. A x sin. \( p \), or sin. \( p (1 - \sin. P \) cof. A) = sin. P sin. A cof. \( p \); hence tang. \( p = \frac{\sin. P \sin. A}{1 - \sin. P \text{cof. } A} \). This formula seems a little difficult to be wrought in numbers, but it is as easy as the above one; for, supposing sin. B = fin. P cof. A, the tables will give the angle B, and tang. \( p = \frac{\sin. P \sin. A}{\cos. B^2} \), the computation of which can give no trouble. Hence it appears, that the calculus for finding the true parallax is not more difficult than that, which gives the said parallax with an error, the value of which is unknown; for it is evident that the above computation for finding \( p'' \) is only an approxi- mation, and that, to make it accurate, it would be ne- cessary to carry it still on by finding \( \sin. p''' = \sin. P \) \( \sin. (A + p'') \), and afterwards \( \sin. p'''' = P \sin. \) \( (A + p'') \) &c. § 3. I therefore think myself in the right to prefer my method to that hitherto used by astronomers. To confirm my opinion, I made a trial, by putting \( P = 59' \) and \( A = 30^\circ \), and found \( p - p'' = 0'',43 \), in which the error of the usual computation amounts to near half a second; I therefore give the preference to the geometrical calculus. § 4. Before I quit the formula tang. \( p = \frac{\sin. P \sin. A}{1 - \sin. P \cos. A} \), I must observe, that the computation of \( p \) may be executed by other methods to the same exactness. If we take \( \cos. 2 C = \sin. P \cos. A \), we shall have tang. \( p = \frac{\sin. P \sin. A}{2 (\sin. C)^2} \), and the computation of this new formula is extremely easy. § 5. The formula tang. \( p = \frac{\sin. P \sin. A}{1 - \sin. P \cos. A} \), gives besides, \( \sin. p = \frac{\sin. P \sin. A}{\sqrt{1 + \sin. P^2 - 2 \sin. P \cos. A}} \); make \( \sin. \) \( P = 2 \cos. D \), \( D \) being a given angle, of which we may have tables ready made, and we shall have \( \sin. \) \[ p = \] \[ p = \frac{\sin P \sin A}{\sqrt{1 + 2 \sin P (\cos D - \cos A)}} = \frac{\sin P \sin A}{\sqrt{1 + 4 \sin P \sin \frac{A+D}{2} \sin \frac{A-D}{2}}} \] \[ \sin \frac{A-D}{2}; \text{ since } \cos D - \cos A = 2 \sin \frac{A+D}{2} \sin \frac{A-D}{2}. \] This being found without any logarithmic computation, we shall find tang. \( E^2 = 4 \sin P \sin \frac{A+D}{2} \sin \frac{A-D}{2} \), if \( A > D \), and hence we may easily compute \[ \sin p = \sin P \sin A \cos E; \text{ but if } A < D \text{ we shall find } \cos F^2 = 4 \sin P \sin \frac{A+D}{2} \sin \frac{A-D}{2} \text{ and hence } \sin p = \frac{\sin P \sin A}{\sin F}. \] § 6. Similar formulae may be found for \( \cos p \), but as the angle \( p \) is pretty small, one might easily fall into some error by the usual tables of logarithms. I shall not say what would be the amount of this error of \( p \), having furnished the manner of avoiding it; but this remark has not, I think, as yet been made in astronomical calculations; and I have found it of great consequence in computing eclipses, where the distances to be found are very small arches. § 7. It may moreover be observed, that if \( A = D \), \[ \sin p = \sin P \sin A; \text{ hence } p' = p \text{ in the same case, and } p'' > p, \text{ which seems very odd; but the moon then is below the sensible horizon. } \] **Theory of the apparent Diameters of the Moon.** § 1. First the expression of horizontal diameter of the moon, or of the diameter seen at the horizon, seems to me too vague; for one ought to understand by it the diameter seen at the center of the terrestrial terrestrial sphere, rather than the apparent diameter at the horizon, which is not affected by refraction. Without this, if the one was confounded with the other, an error would arise for the latitude of Paris from $0''_{25}$ to $0''_{32}$. § 2. Let us keep the same denominations of $P$, $p$, and $A$, and call $D$ the apparent semi-diameter of the moon at the centre of the sphere, and $d$ the apparent semi-diameter of the moon at the zenith distance $= A$. We shall have $\sin A : \sin A + p :: \tan D : \tan d$, or if one will, $\sin A : \sin A + p :: D : d$: the error not exceeding an 100th part of a second. § 3. We had above $\sin p = \sin P \sin A + p$. Hence $\sin P \sin A : \sin p :: (\tan D : \tan d) :: D : d$, or because $\sin p = \frac{\cos p \sin P \sin A}{1 - \sin P \cos A}$, $1 - \sin P \cos A : \cos p :: D, d$, and $d = \frac{D \cos p}{1 - \sin P \cos A}$. § 4. Mr. Euler, in the Memoirs of the Academy of Berlin, 1747, pag. 175, makes this same value $= \frac{V}{1 - p^2 \sin b}$, and according to him, $V = D \cdot M = \sin P \sin b = \cos A + p$; from whence it appears, that the true value of the apparent diameter of the moon, is not more difficult to be computed than the approximated one of Mr. Euler, the exact and geometrical formula being $\tan d = \frac{\tan D \cos p}{1 - \sin P \cos A}$ and that of Mr. Euler $d = \frac{D}{1 - \sin P \cos A + p}$; for in both, the values of $D$, $A$ and $p$ must be employed. § 5. § 5. It likewise appears to me, that since \[ \frac{\cos. p}{1 - \sin. P \cos. A} = \frac{\sin. p}{\sin. P \sin. A} \] and therefore \[ \tan. d = \frac{\tan. D \sin. p}{\sin. P \sin. A}, \] astronomers ought no less to employ this last formula, than any other more troublesome, in practical computation. The simplest is \[ \tan. d = \frac{\tan. D \sin. A + p}{\sin. A}, \] upon the supposition of an exact table of the parallaxes of altitudes ready made; and I believe it will be as easy to compute with tangents as with arches, by means of logarithms; and therefore this simplification in putting arches instead of tangents is unnecessary. § 6. To try the consequences of this theory, I made \( A = 30^\circ, D = 15' \), and taking the vertical of Upsal to the terrestrial axis for the radius of the sphere, I found \( P = 55', 10'', 3 \), supposing that the axis of the earth, is to the diameter of the equator as 199 to 200, and by the formulæ \[ \tan. d = \frac{\tan. D \sin. A + p}{\sin. A} = \frac{\tan. D \cos. p}{1 - \sin. P \cos. A} = \frac{\tan. D \sin. p}{\sin. P \sin. A}, \] I found \( d = 15', 12''.664 \), but by the formula \[ d = \frac{D \cos. p}{1 - \sin. P \cos. A}, \] I had \( d = 15', 12'', 675 \), and lastly by that of Euler \[ d = \frac{D}{1 - \sin. P \cos. A + p}, \] we have \( d = 15', 12'', 635 \); from whence it appears that the error is very small, but that with the same trouble one may avoid any error whatsoever. § 7. The present case did not give an error of 0'', 001 in substituting 1 or the radius instead of cos. \( p \). Hence I conclude that \[ d = \frac{D}{1 - \sin. P \cos. A} \] will be a more a more exact formula than that of Euler \( d = \frac{D}{1 - \sin P \cos A + p} \). § 8. By taking \( d = \frac{D}{1 - \sin P \cos A} \), we have \( d - D = \frac{D \sin P \cos A}{1 - \sin P \cos A} = \frac{D \tan p}{\tan A} \), which affords an elegant theorem, to find the increase of the apparent diameter of the moon. § 9. I have found others by the following methods. Since \( \sin A : \sin A + p :: \tan D : \tan d \), and \( \sin A : \sin A + p - \sin A :: \tan D : \tan d - \tan D :: \sin D \cos d : \sin d - D \); but \( \cos D = \cos d \) without any sensible error, and \( \sin D \cos D = \frac{1}{2} \sin 2D \), and \( \sin A + p - \sin A = 2 \sin \frac{1}{2} p \cos A + \frac{1}{2} p \), we shall have \( \sin d - D = \frac{\sin 2D \sin \frac{1}{2} p \cos A + \frac{1}{2} p}{\sin A} \). In the same manner, as I before found \( \sin p' = \sin P \sin A \) and \( \sin P \sin A : \sin p :: \tan D : \tan d \), hence \( \sin p' : \sin p - \sin p' :: \sin p' : 2 \sin \frac{p - p'}{2} \cos \frac{p + p'}{2} :: \frac{1}{2} \sin 2D : \sin d - D = \frac{\sin 2D \sin \frac{p - p'}{2} \cos \frac{p + p'}{2}}{\sin P \sin A} \). § 10. Lastly let \( L = \) the distance of the moon from the center of the sphere, \( l \) its radius, that of the sphere being \( = 1 \), we have \( 1 : L :: \sin P : 1 \) and \( L : l :: 1 : \tan D \) or \( 1 : l :: \sin P : \tan D = l \sin P \); hence \( l = \frac{\tan D}{\sin P} \) being once found, since \( \sin A : \sin A + p :: \tan D : \tan d \), and \( \sin A + p : \sin p :: 1 : \sin P \), we shall have \( \sin A : \sin p :: \tan D : \sin P \tan d :: l : \tan d = \frac{l \sin p}{\sin A} \). I found the logarithm of \( l = \) 9.4343965 at Upsal, by putting 10 for that of the radius of the sphere determined as before. F. Mallet. XXX. A Catalogue of the Fifty Plants from Chelsea Garden, presented to the Royal Society by the worshipful Company of Apothecaries, for the Year 1765, pursuant to the Direction of Sir Hans Sloane, Bart. Med. Reg. et Soc. Reg. nuper Praeses: By William Hudson, Societatis Regiae & clariss. Societatis Pharmaceut. Lond. Soc. Hort. Chelsean. Praefectus et Praelector Botanic. Read Nov. 20, 2151 A CHYRANTHES lanata, caule prostrato, spicis ovatis lateralibus, calycibus tomentosis. Lin. Sp. pl. 296. Mill. Dict. tab. 11. fig. 1. Amaranthus Indicus verticillatus albus, foliis lanugine incanis. Pluk. alm. 27. tab. 79. f. 8. 2152 Andrachne procumbens herbacea. Lin. Sp. pl. 1439. Telephoides Graecum humifusum flore albo. Tourn. cor. 50. Dill. Hort. Elth. 377. tab. 282 f. 364. 2153 Bryonia Africana, foliis palmatis quinquepartitis utrinque laevibus: laciniis pinnatifidis. Lin. Sp. pl. 1438. Bryonia