VARIATION. 



VaUIATION or Declination of the Needle, To ohferve the 



Draw a meridian line, as direfted under Meridian, then, a 

 ilyle being erefted in the middle of it, place a needle thereon, 

 and draw the right line which it hangs over. Thus will 

 the quantity of the variation appear. 



Or thus : as the former method of finding the declination 

 cannot be applied at fea, others have been thought of, the 

 principal of which follow : fufpend a thread and plumbet 

 over the compafs, till the fliadow pafs through the centre 

 of the card ; obferve the rhumb, or point of the compafs, 

 which the (hadow touches when it is the Ihorteft. For the 

 fhadow is then a meridian line ; confequently the variation 

 is (hewn. 



Or thus : obferve the rhumb in which the fun, or fome 

 ftir, rifes and fets ; bifedt the arc intercepted between the 

 rilmg and fetting ; the line of bifefting will be the meridian 

 line ; confequently the declination is had as before. The 

 fame may be had from two equal altitudes of the fame ilar, 

 obferved either by day or night. 



Or thus : obferve the rhumb in which the fun, or a (lar, 

 rifes and fets ; and from the latitude of the place find the 

 eaftern or weftern amplitude, for the difference between the 

 amplitude, and the diftance of the rhumb obferved, from the 

 eaftern rhumb of the card, is the variation fought. 



Or thus : obferve the altitude of the fun, or fome ftar, 

 SI [Plate \\. Navigation, fg."].) whofe declination is 

 known ; and note the rhumb in the compafs to which it 

 then correfponds. Since then, in the triangle Z P S, we 

 have three fides ; viz. P Z, the complement of the elevation 

 of the pole PR; S P, the complement of the declination 

 D S ; and Z S, the complement of the altitude SI; the 

 ano-le P Z S is found by fpherical trigonometry ; the con- 

 tiguous one to which, -viz. A Z S, raeafures the azimuth H I. 

 The difference, then, between the azimuth, and the diftance 

 of the rhumb obferved from the fouth, is the variation 

 fought. See Azimuth Compass. 



Note, to have the eaftern or weftern amplitude accu- 

 rately, regard muft be had to the refraftion. See Re- 

 fraction. 



For the more commodious obferving in what rhumb of 

 the compafs the fun, or a ftar, is feen, it will be proper to 

 have two little apertures, or glafs windows, oppofite to each 

 other, under the liml of it, with a telefcope-fight fitted 

 to one of them, and to the other a fine thread. 



The ufe of the variation is to correft the courfes a fhip 

 lias fteered by the compafs : e. g. given the courfe fet, and 

 the variation of tlie compafs, and let it be required to find 

 the true courfe the fliip fteers : if the variation is weft, call 

 the N.W. quarter the ift, the S.W. the 2d, the S.E. the 

 ■^i, and the N.E. the 4th ; but if the variation be eaft, 

 call tlie N.E. quarter the ift, the S.E. the 2d, the S.W. 

 the 3d, and the N.W. the 4th. Then, if the courfe be fet 

 in the i ft or 3d quarters, add the variation to the points or 

 degrees in the given courfe ; but if in the 2d or 4th quarters, 

 fubtraft, and the fum in the former, or the difference in the 

 latter cafe, will be the courfe correfted by the variation. 



Variation of Curvature, in Geometry, is ufed for that 

 inequability, or change, which happens in the curvature 

 of all curves, except the circle. And this variation or ni- 

 equability conftitutes the quality of the curvature of 

 any line. 



Sir Ifaac Newton makes the index of the inequability or 

 variation of curvature, to be the ratio of the fluxion of the 

 radius of curvature to the fluxion of the curve : and Mr. 

 Maclaurin, to avoid the perplexity that different notions, 

 conneded with the fame terms, occafion to learners, has 

 adopted the fame definition ; but he fuggefts, that this ratio 



gives rather the variation of the ray of curvature, and that it 

 might have been proper to have meafured the variation of 

 curvature rather by the ratio of the fluxion of the curvature 

 itfelf to the fluxion of the curve ; fo that the curvature being 

 inverfely as the radius of curvature, and, confequently, its 

 fluxion as the fluxion of the radius itfelf direftly, and the 

 fquare of the radius inverfely, its variation would have been 

 direftly as the meafure of it, according to fir Ifaac Newton's 

 definition, and inverfely as the fquare of the radius of 

 curvature. 



According to this notion, it would have been meafured 

 by the angle of contaft contained by the curve and circle 

 of curvature, in the fame manner as the curvature itfelf is 

 meafured by the angle of contaft contained by the curve and 

 tangent. The reafon of this remark may appear from this 

 example. The variation of curvature, according to (ir 

 Ifaac Newton's explication, is uniform in the logarithmic 

 fpiral, the fluxion of the radius of curvature in this figure 

 bemg always in the fame ratio to the fluxion of the curve ; 

 and yet, while the fpiral is produced, though its curvature 

 decreafes, it never vanifhes, which muft appear a ftrange 

 paradox to thofe who do not attend to the import of fir 

 Ifaac's definition. Newton's Meth. of Flux, and Inf. 

 Series, p. 76. Maclaurin's Fluxions, art. 386. Phil. 

 Tranf. N° 468. feft. 6. p. 342. 



The variation of curvature at any point of a conic fcc- 

 tion, is always as the tangent of the angle contained by the 

 diameter that paifes through the point of contaft, and the 

 perpendicular to the curve at the fame point, or to the 

 angle formed by the diameter of the feftion, and of the 

 circle of curvature. Hence the variation of curvature 

 vanifhes at the extremities of either axis, and is greateft 

 when the acute angle, contained by the diameter, palfing 

 through the point of contaft and the tangent, is leaft. 



When the conic feftion is a parabola, the variation is as 

 the tangent of the angle, contained by the right line drawn 

 from the point of contaft to the focus, and the perpen- 

 dicular to the curve. See Curvature. 



Variation of Ratios. In the inveftigations of the re- 

 lation which varying and dependent quantities bear to each 

 other, conclufions are frequently more readily obtained by 

 expreffing only two terms in each proportion, than by re- 

 taining the four. But although in confidering the variation 

 of fuch quantities two terms only are expreffed, it will be 

 neceffary to boar conftantly in mind that four are fuppofed, 

 and that the operations by which our conclufions are in this 

 cafe obtained, are in reaUty the operations of four pro- 

 portionals. 



1. One quantity is faid to vary direBly as another, when 

 their magnitudes depend wholly upon each other, and in 

 fuch a manner, that if the one be changed, the other is 

 changed in the fame proportion : thus, let A and B be 

 mutually dependent upon each other in fuch a way, that if 

 A changes to any other value a, B is chang;ed to another 

 value b, fuch that A : a :: B : A ; then A is faid to vary 

 direftly as B, which is denoted by the fymbol of general 

 proportion oc placed between the two quantities. Thus, 

 for example, while the altitude of a triangle remains con- 

 ftant, the area varies direftly as the bafe, or the area oc bafe ; 

 for if the bafe be incrcafed or diminiflied, the area is in- 

 crealed or diminifhed in the fame proportion. 



2. One quantity is faid to vary inverfely as another, 

 when one cannot be changed in any manner ; but the reci- 

 procal of the other is changed in the fame proportion. 



A varies inverfely as B, or A oc -^, if when A is changed 



to 



