ALT 



Radius 



Log. of T) C, or 64 feet 

 Tang, of ji'^ 52' 



lO.OOCOOO 



1. 8061 80 

 10.105108 



1. 9 1 1280 



.ACor8ilftet 

 To which add four feet, the Iieiglit of the eye, and the 

 altitude required, or A B, is 85} feet. 



This may alio be refolved by projedion, thus : draw DC, 

 on which fet 64 feet from any feale from D to C ; ereft tlid 

 perpendicular CA ; then make the angle CD A = 51^ 52', 

 and draw DA, interfefting the perpendicular in A, tiic top 

 of the objeft. Then CA, meafured ontliefarr.e feale, will 

 give 8i§. 



If there happen an error in taking the quantity of the 

 angle A, {fig. 6.) the true altitude BD will be to the falfe 

 one BC, as the tangent of the true angle DA B, to the 

 tangent of the erroneous angle CAB. 



Hence, fueh error will be greater in a greater altitude 

 than in a lefs ; and hence alfo, the error is greater, if the 

 -angle be leffer, than if it be greater. To avoid the incon- 

 veniences of both which, the ftation is to be pitched on at 

 a moderate dillance ; fo that the angle of altitude DAB, 

 mav be ncarlv half right. 



Again, if the inftrunicnt were not horizontally placed, 

 but inclined, e. gr. to the horizon in any angle, the true 

 altitude will be to the erroneous one, as the tangent of the 

 true angle to that of the erroneous one. 



If the plane interpafed between the obferver, and the 

 objeft be inclined, as infig. 7 ; two ftations C and D mud 

 be felefted, and ihtir d'llances from the bafe of the objeft, 

 •viz. CA and DA muft be meafured. Then as the ey.ternal 

 angle ACB is equal to CD B + DBC, the angle DBC = 

 ACB - CDB, or DC Bis equal to the fupplement of 

 ACB ; in either way the angles of the triangle BCD are 

 known, and one fide DC is given; then fay, the fine of 

 DBC : DC : : fine of BDC : BC, which will be 

 known : and in the triangle ABC, the two iides CA and 

 CB being given together with the included angle, we fliall 



B + A 

 have CB+CA : CB - CA : : tangent of : 



tangent of 



B 



", whence the angles will become known ; 



and it will be eafy to find AB the altitude of the objeft as 

 before. Otherwife, meafure the dillance AC, and the 

 angles AandC ; and as in the trianglcACB, all the angles and 

 one fide AC are given, the other fide A B will be eafily 

 found. 



To meafure an accefiible aUitude optically, by the fhadow 

 of the body, fee Shadow. 



To meafure the ahitude of any objeft by optical refleftion, 

 place a plane mirror, or a vefTel of clear water, horizontally 

 at C, (^fg. 8.), and retreat from it to fuch a diftance at D, 

 that the eye E mav juft perceive the image of the top of the 

 objtft, in the reflefting furface at C ; then, as theic triangles, 

 having two equal right angles, and the angle ACB=:ECD, 

 becaufe the angle of incidence is equal to the angle of re- 

 fleftion, aie fimilar, we fliall have CD : DE : : C A : AB, 

 the altitude required. 



To meafure an acceffible altitude by the geometrical qua- 

 drat or fquare. Suppofe it required to find the ahitude 

 AB {fg. 9.) choofing a ftation at pleafure in D, and mea- 

 furing the diftance thereof from the objeft DB ; turn the 

 quadrat this and that way, till the top of the tower A appear 

 through the fights. 



If then, the thread cut the right (hadows, fay, as the 



ALT 



part of thff right fliadow cut off, is to the fide of the qua- 

 drat, fo is the diftance of tlic flation DR, 10 the pan of the 

 altitude A E. If the thread cut the vcrftd Ihaduw, fay, »■ 

 the fide of the quadrat ia to the part of the vcrfcd flindow 

 cut off, fo ib tile diHance uf llic Aation DB, tu the part uf 

 the altitude A E. 



A E, therefore, bcinjj found in either cafe, by the rule of 

 three, and the part of the altitude BE added to it, the fura 

 is the altitude required. See Qi'adrat. 



Ai.TiTi'DE, to meafure an inaceeffible, gcvmelritnlly, Sup- 

 pofe A B {^fig. 4.) an inaceefTiblr altitude, fo that you cannot 

 meafure to the foot of it. Find the diilanec C A, or Fll, 

 as taught under the article Distance ; then proceed with 

 the reil as in the article for aceeflible ddlances. See Staff. 



To mealuie an iiiaceclTible allitiiJe, Irigonofnetrically.— 

 Choofe two nations G and E (Jig. 10.) in the fame right 

 line with the required altitude A B, and at fuch diflancc from 

 each other, DE, as- that neither the angle EAD be too 

 fmall, nor the other ftation G too near the objeft AB. With 

 a proper inftrumeiit take the quantity of the angles ADC, 

 A EC, and C F B ; and alfo meafure tile interval F D. 



Then, in the triangle A FD, we liave tlic angle D, given 

 by obfervation ; and llie angle A FD, by fubt rafting the 

 obfeived altitude AFC, from two right angles ; and eon- 

 fequently ihethird angle 1) A 1", liy fubt ratting the other two 

 from two right ones; and allbtlie fide FD; from whence the 

 fide A F is found by the canon above laid down, in th<' 

 problem of aeceffible altitudes. And again, in the triangle 

 ACF", having a right angle C, and oblcned angle 1', and 

 a fide AF, the fide AC, and the other CF, are found by 

 the fame cnnon. LalUy, in the triangle FCB, having a 

 right angle C, obferved angle C Y B, and a fide C 1' ; the 

 other fide CB, is found by the fame canon. 



Adding, therefore, AC, and CB, the fum is the altitude 

 required, AB. 



E. G. Suppofe AFC to be 58° and ADC 38'', and 

 the diftance of the ftations FD to be 26 yards. Subtrart 

 ADC or 38" from AFC or 58^ and there remains FAD 

 or 20°. Then, in the triangle DAF, the angles and one fide 

 beinjr known, we fliall have fine of DAF . fine of ADl' 

 : : FD : FA /. c. S. 20° : S. 38° : : 26 : a fourth, or 

 by logarithms, 9.5340517 : 9.7893420 : : i. 414973 : 



9.7S93420 4- 1-414973 - 9-5.^40917 = 1-6:02633 the 

 log. of 46.8. Again, in the triangle A F C, radius : fine 

 of AFC : : AF : AC, /. t. rad. : S. 58° : : 46.8 : 

 a fourth, and by logarithms, lo.ooocooo : 9.92S4205 : : 

 1.6702633 : 9.9284205 -f- 1.414973 — 10.0000000 = 

 1.5986838 the log. of 39.69 or 39 yards two feet, to which 

 add the height of the inftrumcnt above the ground, and we 

 have the altitude required. 



By projeftion ; draw a line DC, and at the extremity D 

 make the angle ADC = 38'^, and draw thehne D A. Then 

 fet oft' the diftance of the ftations 26 from D to F, and at 

 1" make the angle AFC = 58", and draw FA to intcrfrft 

 DA in A : then the diftance cf A from the horizontal hne 

 DC, applied to the fcate, will give the height. 



If two ftations be taken at F and D, L that the angle 

 AFC may be = 2 A D C, D F will be = A F ; and radius 

 will be to the diftance of the ftations DF or A F : : S. 

 AFC : the altitude AC. 



If the ftation at F be fuch, that the angle AFC may 

 be 45'', and the angle ADC = 26° 34', the ahitude AC 

 will be equal to D F the diftance of the two ftations. For 

 when AFC is 45°, AC = CF = FD, and DC = 2 AC, 

 as radius : natural tangent of lO^ 34' : = * '• 1, i- r. : ; 

 DC (or 2 AC) : AC. ■ 



We 



