702 



Theory 

 and Con- 

 struction. 



Vertical 



declining 



dials. 



Piate 

 CCXXIX". 



F.g. 5. 



Tan. O = 



kinds, viz. south-east, south-west, north-east, and north- 

 west decliners. 



The declination of any plane, whether vertical or in- 

 clined, is an arch of the horizon intercepted between the 

 plane and the prime vertical ; or it is the ai'ch of the 

 horizon intercepted between the meridian and a vertical 

 plane, which is also perpendicular to the proposed plane. 



The meridian of any dial plane is a plane that passes 

 along the axis, or edge of the stile, and is perpendicu- 

 lar to the plane of the dial. 



The substile of a dial is the common section of its 

 plane, and the plane of its meridian In horizontal, 

 and in vertical south and north dials, the substile coin- 

 cides with the twelve o'clock hour line; but in decli- 

 ning dials this is not the case. 



The difference of longitude of a dial plane is the 

 angle which the plane of its meridian makes with the 

 meridian of the place. 



61. Let PQ, Fig. 5, be a vertical plane, a wall for 

 example, having any aspect ; and let us suppose that 

 upon its face that looks towards the south, an axis or 

 stile OC, has occl\ fixed at O, in a position parallel 

 to the axis of the diurnal mouGH, by what has been 

 taught in art. 32. and 33. 



It appears, in the first place, that the vertical line 

 OB drawn from the point in which the stile meets 

 the wall, will be the 12 o'clock hour line; for it is com- 

 mon to all the vertical planes which pass through C, Cos. (C- 

 and consequently must be the intersection of the plane an( ^ hence, 

 of the dial, and the meridian of the place. 



From C, the extremity of the stile, draw CB per- 

 pendicular to C XII, thus forming the right angled 

 triangle OBC, which will be entirely in the plane of 

 the meridian ; and therefore the prime vertical WOE 

 is perpendicular to it. Let us suppose that at any hour, 

 for example two in the afternoon, the horary plane, 

 (or plane passing through the axis and the sun,) cuts 

 the prime vertical WOE in the direction Oy, and the 

 plane of the dial POQ in the direction OY ; the first of 

 these lines indicates the hour on the prime vertical ; 

 and the second shews it on the plane of the dial ; but 

 to trace the line OY, we must know the angle BOY, 

 and the whole difficulty of constructing the dial lies 

 in the determination of this angle. 



62. Let us suppose a horizontal plane to pass along 

 CB, and meet the horary plane COy in the line CYy ; 

 then it is manifest that BCY may be considered as the 

 plane of a horizontal dial, of which C is the centre, CO 

 the axis, CB the meridian line, and CY the hour line 

 for two in the afternoon ; therefore the angle BCY will 

 be known by formula 1. (art. 39.) And because the 

 horizontal lines BC, By, lie, the one in the meridian, 

 and the other in the prime vertical, they contain a right 

 angle; now the angle YBy is the declination of the 

 plane, (art. 60.) therefore CBY is its complement, and 

 is known, because we suppose the declination known ; 

 hence all the angles of the triangle BCY are known. 



Let the latitude of the place for which the dial is to 

 be constructed be expressed by L, and the angle 

 y BY, or EOQ the declination of the plane by D ; and, 

 as in the formula of art. 39, let the angle C, made by 

 the hour line of a horizontal dial for the latitude L 

 and the meridian line BC be denoted by C, then, in 

 the triangle BCY, we have the angle at C=C, the angle 

 B=90° — D, and therefore the angle BYC=180 — (90 

 — D)_C=90— (C— D.) V 



In the right angled triangles, OBC, OBY, which 

 have OB, one of the sides about the right angles, com- cing, we find, 



DIALLING. 



mon to both, we have, by the principles of trigonome 

 try, 



BC : BY : : tan. BOC, or co-tan. OCB : tan. BOY. 

 But in the triangle BCY, we have also 

 BC : BY ( : : sin. Y : sin. C) : : cos. (C— D) : sin. C. 

 Therefore, 



Cos. (C— D) : sin. C : : co-tan. OCB : : tan. BOY. 

 Now, the three first terms of this proportion are 

 given, because the angles C and D are given, and also 

 the angle OCB, which is the latitude ; therefore, the 

 fourth term, or the tangent BOY is known, and hence 

 the angle BOY itself is known. 



63. From the foregoing investigation, and the formu- 

 la of art. 39, we derive a formula for the construction of 

 a vertical declining dial, which may be expressed thus : 

 Let L=the latitude of the place. . 



D=the declination of the dial, reckoned from the 



east towards the south. 

 E=the hora'ry angle the sun has described since 



noon. 

 0=the angle BOY, which the shadow has describ- 



ed about the centre of the dial since noon. 

 C=the angle which the shadow has described about 

 the centre of a horizontal dial for that place in the same 

 ;-'"ip. and which is found by the formula, 



Tan. C=sin. L tan. E. (art. 39.) 

 then we have 



Theory 

 and Con. 

 struction. 



Vertical 



declining 



dials. 



-D) 



OOt. Li I 



cot. L sin. C 



ni. O. 



(5.) 



cos. (C— D) 

 In the construction from which the preceding formu- 

 la was derived, we considered the half of the plane of 

 the dial, which passed between the meridian and the 

 prime, vertical, and hence we found the angle CBY= 

 90° — D ; however, the formula is general for all hora- 

 ry angles, only, in conformity to the law of geometri- 

 cal continuity, if we regard the values of C for the 

 afternoon as positive, those for the forenoon must be 

 considered as negative ; and as, by the Arithmetic of 

 Sinks, we have — sin. (-fC)=sin.( — C), and cos. ( — C 

 — D ) =cos. (C + D ), the formula for the forenoon hours 

 is, 



„ _. cot. L sin. C. 



Tan. O = m ,Ti\ f 



cos. (C -f D ) 



The negative sign shews that the angle O ought ta 

 be taken on the other side of the meridian. 



64. The above formula, although very simple, and 

 well adapted to calculation, has yet the inconvenience of 

 requiring two operations for each hour line ; viz. one 

 to find C from E, the horary angle, and another to find 

 O from C. It will, therefore, be proper to investigate 

 a formula that shall give the value of O in terms of E 

 at once. 



For this purpose, in the denominator of the formula, 

 instead of cos. (C — D), put its equal cos. C cos. D-J- 

 sin. C sin. D, (Arithmetic of Sines, art. 7.) and af- 

 ter dividing the numerator and denominator by cos. C } 



^ , . r. sin. C 



let tan. C be put for ~, 



* cos. L 



_, _ co- tan. L. tan. C 

 Tan. O = 



and the result will be, 



cos D + sin. D tan. C 

 Now, let sin. L tan. E be substituted for tan. C, (art. 



39. ) and again, — ~~W~ f° r tan - ^ ar * ( ^ * nen ^^ rethi- 



