Theory 

 and Con- 



Plate 

 CCXXIX. 



Fig. 16. 



Fig. 17. 



DIALLING. 709 



A on that day. Therefore if we recur to the notation 



Dials with Variable Centres. employed in the solution of the problem in last article, 



and express the time from noon by q>, then the angle 



83. Dials of this kind, although not common, nor at C will be 6, and so we shall have 

 very convenient, are yet curious, and deserve to be Cos. (e — x) cos. <t> — sin, (t — x) tan. 5 



known, on account of the elegance of their mathemati- ° * — Sin. t sin. <p ' 



cal theory. This may be deduced from the following Draw AB perpendicular to the meridian ; put OBzrr, 



proposition, which seems to be one of the class called and AB=y, (so that x and y are the co-ordinates of any 



by Geometers, Porisms. hour point,) and put v for OC, the variable distance of 



A system of hour points may be found on a plane, the bottom of the stile from the fixed point O, in the 



such, that for every day in the year (or any time in plane of the dial ; then BC= x — v ; and because AB : 



which the sun's declination may be regarded as con- BC : : rad. : cot. C, that is y : x — v : : 1 : cot, 8, we have 

 stant) there is a point in the meridian line at which, if x — •» 



a stile be placed in the plane of the meridian, so as to c °t- ^ ==— ~ » an " 



make with the horizon any given angle, its shadow * 



shall pass through the different hour points at the in- x — w_Cos. (e — x) cos. <p — sin. (s — x) tan. 3 



stants of time corresponding to them, and in this way ~T| Sin. e sin. tp ' 



shew the time of the day. and hence we find, 



In order to investigate the truth of this proposition, p t am £ sm# ^ „ cos> ( 6 x ) cos. <p 



we shall resolve the following problem. V Sin t sin d> 



Problem. Having given the sun's declination, and the v=< ■ i ' ^\ 



time from noon, and also the latitude of a place, to find I -J- -. '—. — -— tan. d. 



the angle which the shadow of a stile makes with the *- „ sm ' £ ??", 



meridian line on a horizontal plane, supposing the stile The nature of the dial requires that the position of 



to lie in the plane of the meridian, and to make with the stile should depend entirely on the sun's dechna- 



the horizon a given angle tion, without any regard to a particular hour of the 



LetLMNbe the horizon, LPN the meridian, Pthe day; and also that the position of the hour points 



pole, CQ the stile, which meets the meridian in the should depend entirely on the hour of the day, with- 



heavens at Q ; let S be the sun in the hour circle PS, out any regard being had to the sun s declination : But 



and MSQX a great circle passing through S and Q, these two conditions will manifestly be satisfied, if, m the 



and cutting the plane of the horizon in the line MCX. la st equation, we make 



Put x for PN, the given latitude ; * sin - e sin - *—■ 9 cos ' ^~ *) cos> ^= ' 



• for QN, the given elevation of the stile ; and at the same time put 

 i for the comp. of PS, the sun's given dec. V sm -_l|— *) =a a constant quantity, 



<p for QPS, the given hour angle ; Sin. i sin. <p 



■vj/ for the angle LQM ; for then we shall have x and y both independent of 3, 



f for the angle XCN or arc LM, which is to be the declination, and v—a tan. S a quantity independent 



\ found. of <p, as they ought to be. 

 By Spherical Trigonometry, if a and o be the By resolving our two assumed equations in respect of 



sides of a spherical triangle, C the angle they contain, x and y, we readily find 

 and A the angle opposite to a, then _ Cos. (e — x) 



Cot. A sin. C + cos. C cos. i=cot. a sin. b. x ~ a Sin. {i—xf° S ' * ^ > 



Also in a right angled spherical triangle, if a and b Sin. t 



be the sides about the right angle, and A the angle op- ^ =fl SnTT7— xf^' ® ^ 



posite to a, . * V ' /n\ 



r " n . ■ , - . A v=a tan. d (C) 



Cot. a sin. 6=cot. A. _, , , -. , 



In the first of these two formulas, let PS=90— *, and Th ese three equations express completely the nature 



PQ= £ — X be put for a and b; also P=<p for C, and of every dial of this kind, and are sufficient for its con- 



SQP= 1 8Co ^ for A and we get struction, nothing more being necessary than to assume 



—cot. * sin. 9 + cos.' (_ x) cos. <p=sin. (•— >) tan. 2; for « a line of any length whatever, as a scale on which 



and in the second let sin. QL=sin. i be put for sin. b, the parts of the dial are to be measured ; then to com- 



and LM=0 for a, also LQM=J, for A, and the result P ut e x and y from the formulas (A) (B) by making 



^.jjUjg ^ T <p=15°for the hour lines of XI and I, and ?> = 30 for 



Cot 6 sin «=cot -l t ^ 103e of X and ^' and so on ' and lastlv > to form a 



Let this value of cot. * be substituted instead of it in graduated scale along the meridian line, proceeding 



the preceding equation, and we shall get, by transposi- hoth ways from O that point being the position of the 



tion and division st »e at the time oi either equinox : And as, by con- 



r . . Cos. ( £ _x) cos. <p_sin. (_x) tan. 3 siderin § Fi ^, 34 » { \ will readily appeal- that for any 



Cot. 0= ^ — -. : ^ '- » given hour the angle 0, (or C on the dial,) ought to 



sin. i sin. <p increase as the sun approaches the north pole ; the 



From this formula, we may find the value of 0, the scale of declination for the north side of the equator 



angle made by the shadow and the meridian, as re- must lie on the north side of O, and that for the south 



quired. side of the equator on the south side of O. The months 



8*. Let us now suppose that Fig. 17 represents any dial and days of the year ought also to be placed on the 



with a moveable centre, O XII being the meridian line; scale opposite to the degrees of declination to which 



XI, X, &c. the forenoon hour points ; and I, II, &c. they correspond. 



the afternoon hour. Also let C be the position of the 85 - From the equations (A) (B), of last artide, we 



bottom of the stile, on any given day ; and A any nn d, 



hour point; join AC, then AC will be the position of x , = a<i cos * — x ) C os.* <p 



the shadow of the stile corresponding to the hour point Sin.' 2 (s — x) 



Theory 

 and Con- 



