. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. wrecked mariners, and travelers similarly situated, have fre-quently resorted to these artifices to obtain their approximate localities. We have frequently remarked in the course of this work, that thebest position of a celestial object, at the time of taking its altitude, forthe purpose of more exactly defining the time, is when the object isnearly east or west; we now propose to show this conclusively, andtherefore give the following INVESTIGATION. To find under w
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. wrecked mariners, and travelers similarly situated, have fre-quently resorted to these artifices to obtain their approximate localities. We have frequently remarked in the course of this work, that thebest position of a celestial object, at the time of taking its altitude, forthe purpose of more exactly defining the time, is when the object isnearly east or west; we now propose to show this conclusively, andtherefore give the following INVESTIGATION. To find under what circumstances, in a given latitude, a small mistake inobserving or correcting the altitude of a celestial object, will produce thesmallest error in the time computed from Z be the zenith, P the pole, r the supposed place, and m the true place of the object. Let ms be a parallel of altitude, join the points m and r, and let pq be the arc of the equator contained between the meridians Pm and Pr. Then as Pm and Pr are equal, mr maybe considered as a small portion of a par-allel of declination rs will be the error in. 230 APPENDIX. altitude, and pq the measure of the required error in time. And as thesides of the triangle msr will necessarily be small, that triangle may beconsidered as a rectilinear one, right angled at s ; and because the anglePrm is also a right angle, the angles smr and PrZ, being each the com-plement of mrs, are equal to each other We now have, rs : mr : : sin. smr (ZrP) : rad. (1) Also, mr : pq : : cos. qr : rad. (2) Multiplying these two proportions together, omiting the commonfactor mr, gives, rs :pq :: cos. qr sin. (ZrP) : (rad.)* (3) But, sin. rP or cos. qr : sin. rZP : sin. ZP : sin. (ZrP) (4)Whence, cos. qr sin. ZrP=sin. rZP sin. ZP (5) The first member of equation (5), is the same as the third term inproportion (3) ,* therefore, proportion (3) may be changed to thefollowing, rs : pq : : sin. rZP sin. ZP : (rad.)2 Whence, pq= ( ^ai£ ) __L_
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Keywords: ., boo, bookcentury1800, booksubjectnavigation, booksubjectsurveying
