S HA
[ 66-]
S H A
dew be cut by a. Plane parallel to the Bafe, the Plane of the Section will be a Circle, and that fo much the lei's, as it is a greater Diitance from the Bafe.
6°- If the Luminous Sphere be lefs than the opake one, the Shadow will be, a truncated Cone: Consequently, it grows ftill wider and wider; and therefore, if cut by a Plane parallel to the Section, that Plane will be a Circle id much the greater as 'tis further from the Bafe,
7°. To find the Length of the Shadow or the Axis of the Shady Cone, projeiled by a lefs opake Sphere, illumined by a larger ; the Diameters of the Two, as C D, and I N ; Tab. Opticks Fig. 12. and the Diitances between their Cen- ters G M being given.
Draw 1' M parallel to CH; then will I M — CF; and therefore FG will be the Difference of the Semi-Diameters G C and I M. Confequently, as F G, the Difference of the Semi-Diameters is to G M, the Diitance of the Centers ; fo is C F, the Semi-Diameter of the opake Sphere, to M H, the Dittance of the Vertex of the fhady Cone, from the Centre of the opake Sphere. If then, the Ratio of I'M to MH be very linall 5 fo that MH and PH do not differ Tery notably, M H may be taken for the Axis of the Shadowy Cone : Other wife the Part P M mutt be fubtrafted from it j which to find, ftek the Arch LK: For, this iubtract- ed from a Quadrant, leaves the Arch I Qj which is the Meafure of the Angle I M P. Since then, in the Triangle M I P, which is rectangular at P ; befides the Angle 1MQ_, we have the Side IM 5 the Side M P is eafily found by plain Trigonometry.
E. gr. If the Semi-Diameter of the Earth M I = 1 ; the Semi-Diameter of the Sun, according to Ricciolus^ will be = 33 5 and therefore G F zsz 32 ; and of Confequence MH— 2i8f: Since then, M P is found by Calculation to bear a very {mail Ratio to M H ; P H is found to be 288t Semi-Diameters of the Earth.
Hence, as the Ratio of the Diftance of the opake Body, from the Luminous Body GM; to the Length of the Sha- ^t- !V1 H, is conflant; if the Diftance be diminifhed, the Length of the Shadow muft be diminifhed likewife. Con- iequently, the Shadow continually decreafes as the opake Body approaches the Luminary.
8°. To find the Length of the Shadow projected by an opake Body, TS, Fig. 13, the Altitude of the Luminary; &, gr. of the Sun above the Horizon, g»a, the Angle SVT, and that of the Body being given. Since, in the Rectangle Triangle S T V, we have given the Angle V, and the Side T S ; the Length of the Shadow T V is had Trigonometry. See Triangle.
Thus, S.tippofe the Altitude of the Sun 37 4c, and the Altitude of a Tower 187 Feet; TV will be found 241I Feet.
9°. The Length of the Shadow T V, and the Height of the opake Body TS, being given; to find the Altitude of the Sun above the Horizon.
Since, in the Rectangle Triangle S T V, the Sides T V andTS, are given; the Angle V is found thus: As the Length of the Shadow T V, is to the Altitude of the opake Body T S, ib is the whole Sine to the Tangent of the Sun's Altitude above the Horizon. Thus, if TS be 30 Foot, and TV 45'; TVS will be found 330.41'.
10°. If the Altitude of the Luminary, JS. gr. The Sun above the Horizon TVS, be 45 . the Length of the Shadow T V is equal to the Height of the opake Body.
1 1<\ The Lengths of the Shadows T Z and T V of the fame opake Body T S, in different Altitudes of the Lumi- nary, are as the Co-tangents of thofe Altitudes.
Hence, as the Co-tangent of a greater Angle is lefs than that of a lefs Angle ; as the Luminary rifes higher, the Shadow, decreafes: Whence it is, that the Meridian Sha- dows are longer in Winter than in Summer.
120. To meafure the Altitude of any Object, E. gr. A Tower A B, (Fig. 14.J by means of its Shadow projected on a Horizontal Plane.
At the Extremity of the Shadow of the Tower C, fix a Stick, and meafure the Length of the Shadow AC; Fix another Stick in the Ground of a known Altitude D E, and meafure the Length of the Shadow thereof E F. Then, as E F is to A C ; ib is D E to A B- If, therefore, A C be 45 Yards, and E D 5 Yards 5 A B will be 327 Yards.
13°. The Shadows of equal opake Bodies have their Length proportionable to their Diflahces from the fame Luminaries equally high. Hence, as the Opake ap- proaches to the Luminary ; or the Luminary to the Opake Body, the Length of the Shado-w is increafed ; and as either of them recedes, is diminifhed. Hence, from the different Lengths of Shadows of the fame opake Bodies at the fame Height of the Sim, Moon, Jupiter* Venus, &c. we may gather their different Diftances from the Earth ; tho' not ac- curately enough for Aftronomical Purpofes, See Dis- tances.
14°. The Right Shadow is to the Height of the opake Bo. dy, as the Co-fine of the Luminary, to theSine,
15 . The Altitude of the Luminary being the fame in both Cafes, the opake Body AC, (Tig. 15.) will be to the verfed Shadow A D, as the right Shadow E B, to its opake Body DB.
Hence, i°. The opake Body is to its verfed Shadow, as the Co-fine of the Altitude ot the Luminary to its Sine; Confequently, the verfed AD is ro its opake Body A C, as the Sine of the Altitude of the Luminary to its Co-fine. 2°. If D B = A C ; then will D B be a" mean Proportional between E B, and ADj that is, the Length of the opake Body is a mean Proportional between its right Shadow and verj'ed Shadcw y under the fame Alti'tude of the Luminary.
■ 3 . When the Angle C is 45° ; the Sine, and Co-fine
are equal; and therefore the verfed Shadow equal to the Length of the opake Body.
15 . A right Sine is to a verfed Sine of the fame opake Body, under the fame Altitude of the Luminary, in a Dupli- cate Ratio of the Co-fine, to the Sine of the Altitude of the Luminary.
Right and verfed Shadows^ are of confiderable Ufe in Meafuring ; as by their Means we can commodioully enough meafure Altitudes, both acceffible and inacceffible, and that too when the Body does not project any Shadow. The right Shadows^ we ufe when the Shadow does not ex- ceed the Altitude; and the verfed Shadows, when the Sha- dow is greater than the Altitude. On this Footing, is made an Inftrument call d the ghtadrat or Line of Shadows^ by means whereof the Ration of the right and verfed Shado-w of any Object, at any Altitude, are determined. This In- ftrument is ulually added on the Face of the Quadrant. Its Defcription and Ufe, fee under the Article Quadrat.
Shadow, in Geography. The Inhabitants of the Globe are divided, with Reipect to their Shadows, into J[fcii t Aniphifcii, Heterofcii , and < Perifcii, The firft are fuch as at a certain Seafon of the Year have no Shadows at all, while the Sun is in the Meridian. See Ascn. The fecond are fiich, whofe Meridian Shadow, at one Seafon of the Year looks to the North, and at another to the South. See Am- phiscii. The third are fuch, whofe Shadows conftamly tend either to the North or South. See Heteroscii. The laft are thofe, whofe Shadows, in one and the fame Day, fuccefftvely tend to all the Points. See Periscii.
Shadow, in Painting, an Imitation of a real Shadow., effected by gradually heightening, and darkening the Colours of fuch Figures as by their Diipofitions cannot receive any direct Rays from the Luminary, fuppofed ro enlighten the Piece. The Management of the Shadows and Lights, makes what the Painters call the Clair obfciire : The Laws whereof fee under the Article Clair-ob<cure.
Shadow, in Perfpective. The Appearance of an opake Body, and a luminous one, whofe Rays diverge, (E. gr, as Candle, Lamp, 0c.) being given; To find the juft Ap- pearance of the Shadow according to the Laws of Perfpective : The Method is this. From the Luminous Body, which is here confidered as a Point, let fall a Perpendicular to the Perfpective Plane or Table; ;. e. Find the Appearance of a Point upon which a Perpendicular drawn from the Middle of the Luminary, falls on the Perfpective Plane ; and from the feveral Angles, or rais'd Points of the Body let fall perpen- dicular to the Plane. Thefe Points whereon thefe Perpendi- culars fall, connect by right Lines, with the Point upon which the Perpendicular let fall from the Luminary, falls. And continue the Lines to the Side oppofite to the Lumina- ry. Laftly, thro* the railed Points, draw Lines thro' the Centre of the Luminary, interfering the former ; the Points of Interferon are the Terms or Bounds of the Shadow,
E. gr. Suppofe it required to project the Appearance of the Skadw>of& Prifm.ABCFED (Tab. <PerJpeclive Fig.%.) Scenographically delineated: Since AD, B E, and C F, are perpendicular to the Plane, and L M is likewife perpen- dicular to the fame ; (for the Luminary is given, if its Atti- tude L M be given) Draw the right Lines G M and H M, thro' the Points D and E. Thro' the raifed Points A and B, draw the right Line G L and H L, interjecting the former on G and H. Since the Shadow of the right Line A D ter- minates in G ; and the Shadow of the right Line B E in H; anrj the Shadows of all the other right Lines conceived i" the given Prifm are comprehended within thefe Terms ; G D E M will be the Appearance of the Shadow projected by the Prifm.
SHAFT, in Building ; the Shaft of a Column, is the Body thereof; thuscall'd from its Streightnefs : but by Architects more frequently the Fuji-. See Fust. Shaft, is aifo ufea for theSpire of a Church Steeple. See Spire. And for the. Tunnel of a Chimney. See Chimney.
Shaft of a Mine, is the hollow Entrance into a Mine, which is funk or dug to come at the Ore, In the Tin-Mines, after this is funk about a Fathom, they leave a little, lon£, iquare Place, which is called a Shamble.
SHAGREEN. See Chacreen.
SHALLOPor SLOOP, is a fmall'light Veffel.with only* fmall Main-mart, and Fore-malt, and Lug-Sails, to hale up.
