Some Considerations on a Late Treatise Intituled, A New Set of Logarithmic Solar Tables, &c. Intended for a More Commodious Method of Finding the Latitude at Sea, by Two Observations

Some Considerations on a Late Treatise Intituled, A New Set of Logarithmic Solar Tables, &c.Intended for a More Commodious Method of Finding the Latitude at Sea, by TwoObservations of the Sun; By H. Pemberton, M. D. R. S. Lond. et R. A. Berol. S.Author(s): H. PembertonSource: Philosophical Transactions (1683-1775), Vol. 51 (1759 - 1760), pp. 910-929Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/105427 .

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LXXXI Soffle Confideratiors on a late I4reif irtituled, A new Set of Logarithmic Solar Tables, 89s. irtended for a fnore corsano- dio?Xs Method af f#@ig tbe Latitade at Sea, by 1Jwo ObJ¢ervations of the Sun; bv H. Perzberton, M. D. R S. Lond. et R. iM. 13erol. S.

Reads6ov.2o, 2 s it happens rlot unfrequently at

1 X fea for the unfeafonable nterven- tion of clouds to prevent the ordinary method of de- ternlining the ffiip's latitude by tlae fun's lneridia altitude, even when it is of primary corlfequence that the true latitude {hould be known; a prt3blem for remedying this difappoilztment is liated in many treatiSes of navlgatlon, for finding t}we l$tltude of a place by any two: altitudes of the fun, with the inter_ val of time between thems

A prQblemg fiumilar to thes is propofed and folved inllrumentally upon a globe, by a very early writer3

Petrus Nonius (a), namelyy to ISnd the latitude by two altits of the fun) and the angle rnade by the aziEnuth circles pa^ng through the fun, srhen tlle altitudes are ta;ken. And fince more £ommodious and accurate infirum¢nts for meafuring tinle have been invented, than were known tc) thLs author, tlle oWher problem has been propofed for the fanze pur-

. . .

(a) Itl Libr De Obire. RegulQ st lnfiru. Geometr. li5. iie

s 14.

pQfj

[ 9IO 3



[ 91t ] poSe, of which a conItrudion upon the principles of the ilereographic ploJedion of the fphere is exhibited by Mr. ( ollisls, in his Mariner's Plain Scale new planed (,). And as the dired method of folving both thefe probletns b)J nurnbers requires a diverf1ty of tri- gononietrical operations, a fet of talDles has lately been publRled for a tnore compendious way of cotnputa tion in the problem, wllere tlie interval of tinle is given, whereby the fbip's true latitude may be very expeditioully derived fiom the fhip's dead reckosling) provided the oltervations are made witlain crtaill li- mits of tinze.

But however wortlay of rlotice this method may be, new tables for tlae purpofe are altogether 1lnlle- ceSary It confiRs of tstro parts: tlle firk conlputes,, froln the latitude ex;liibited by the dcad reckoning of the Illip, the diItancq from lloon of the middle time between the obServations, and thence tbe time of cither: trhe fecond operation computes, from one of thefe obServations, what nzould be the fun's- meri dian altitude, had the {hip's reckoning given thc true latitude; but if the latitude aXumed from that reckon-. i.ng is erroneous, the altitude thus (:omputed will not he conformable to it; however, if the times for- the obServations are properly chofen9 it will much better agree to the true latitude, and thereby the aXumed latitude may be more or lefs correfted.

l3trt lrotll tilesne opel-atiolls are an immediate con fequence fiotn tllLe propofition in fpherical trigono * nzetry, ulually delivered under the name of the fourth axiom, wilich is thi>; mThat the fquare of the raclius is to tlle redangle under t}>£e illaes c)f the fides con

. . . . . . W .

(&) Bookiii! p ,:)). taiili jig



[ 9t2 ] taining any angle, as tlle verfed fine of that angle to the differetice between the verfed fines of the third Iide, and of the difference between the fi1des contais- taining the angle

C F C F _ 1o

Hence if A be the -senlth of any place; B the elevated pole; C} D two places of the fun in the parallel of declination D F; by this propofiton in the trf angXe A B Ca

n KX

-



E 9I3 ] Rad.q : ffn. AB x flll. BC (cof. lat. x cof decl. >))

;; verf. f. ABC : verf, f. AC verf f. AF,

And, in like manner, in the triailgle A B D, llladBq : cof. lat. x cof. decle (i) ; v verf. f. A B D : verfi f. A D - verf. f. AF

Confequentlya Rail.q: corlataXcof.decl s: ;ver.f. ABD-verf.f. ABC : verf. r. AD velf. f. AC.

_ \

]3ut > lad. x veri;f. ABL)-verl.i. ABC is _

ABD + AB(: ABD ABC: -Iin. - x fin z -4 2 -i

Therelires '

Radq : cof. lat. x cof. decl. >' :: fin + f ABD-ABC 1 ra*XYerf. AD--verf.{;AC.

2,

And this is the firk operation in the treatife, thefe re- tuarksconcern. For ill thereEangleXtheIln. +

tlle fin A 3 D-A 3 C, vl?e firflc fideZ when the obSer; vatioas are both made on tlle fatne fide of noon) alld the fccond fitle of this reAangleX wllen one otServaS tion is before noonX and the other after) is tl-le fine of the diAance of the tniddle time between the ob- fervatiolls frornnoon, and the otller 1de is the fine of half the diRance between the obServations. And in the new eables o{le columa exhibits the arithme tical complement of the logarithrllic fine of lla?f the elapfed time, and the next, the logarithm of ttwice the fine of the middle time.

,

But moreover, 2 rad. x s . s. a D - Vv1 t 4. ts w . AD f\C AD -- AC ilng equlal to fin.. Z - X fin. > X tIze

operation may be expreflRed more commodiouily ttlus, Rad*.



[ 9t4 ] RadOt: ceflat xcof.dec-}. C) :: fiin.ABD+ABC

ABD - ABC AD+AC AD - AC ae fin. . : fil. 1 X fin. Z 2 2 Z

xlotsr requiring t}ae common logarlthmic filles only. Again, the fquare of the radius being, as above,

to cof. lat. x cof. decl. O, as the verf. f. A13C to verf.f. A-C_nrerE. f. AF, oras verE.f. ABD to rerf. f. A D-verf. f. A F, tlleSc analogies expreEs the Second operation, which9 in tllis treatlfe, is unnecef fiarily confined to the lefler angle ABC. lnhe co lumn in the table intituled Rif1ng confiRs of loga_ rithmic lrerfed fines, which tnay be applied to either of the angles ABC or ABD promiScuouny; for here AF being equal to the excefs of BC or BD above the arch B A, aXutned for the complement of tlle latitude, the arches AC and AD, in the pre- c.edirlg analogies, will be the complements of the alti_ tudes obferved, if the latitude were truly affiumed, otherwife not; but the difference of their verfed finesX will however be equal to the diffierence of the verfed fines of the cotuplements of the true latitudes; for this is fuppoSe(l in the firll operation. Therefore, if one of the fourth terms of thefe analogies be de- duEed from the lrerEed flne of the complement of the greater altitude, or the other from the verfed fine of the complement of the lefler, the remainder will be the fame, and exhibit a verSed fine for the cotn- plement of the furls altitude different from AF, when B A is affiumed diffeleIlt from the true latitude, and nearer to the truth, if the times for the obServa_ tlons are properly claoSen.

But farther, the two preceding analogies may be reduced to theSe3

- Rad.



[ 9I 5 ]

Rad.9: cof. lat. x cof. decl. 9 :; fin. ABC15

: ^ rad. x verf f. A C rerf. f. A Fa

And Rad.q : cof: lat. x cof. decl. @ *: f1n. ; AB9lt * X rad. x verf.f. AC-verf.f. AF;

Alfo, Verf. f. lNC-verf. f. AF = cof. AF-cof. AC8, and verf.f. AD-verf. f. AF _ cof. AF cof. AD.

And thus the commotl tables of fines will fupply the uSe of this column of the new tables, as sell as the preceding. The firft example of thts treatife, wrought by the

common tables. Here the declination of the fun is ilated at X 1 ° I 7t

N. the altitude firll taken, being the lelIer, 46° S3^. the greater alatude y 7', and the difference of time between tlle cxbServatlons lh 2Sm, or 85nz; whichJ divided by 4, bives 2 I ° x5' for the arch of the tus nodrial correfponding.

SIere the obServations are both before noon; there_ fore, in fig. X . the latitude being aSumed 46 50',

For the firk operation.

A C-3s S3 ° AD- 43 5 °

Their fum _ 78 s8 o Their difference- 7 I2 o

Sineof fum (39 29 a) g.80336 Slneof2 dfference ( 3 36 o) 8.79789

Arith com. of the fine of o C B D (= JO 37 30) o.73+29 Arith. com. of the fine of B F or the cof. of the decl. ] O oo848

sf the fun, - * - - - - - - - - - - ) Salm x

cOf. of thc lat. atIwumed g*83Sr3

- Diff, _fin. A BD + B g.So889

Vo. Ll. 6B Far



For the fccolld opeatio¢ o / ft

SED+Ai3 = 18 49 5°

A B D A B C

_ - _ I 0 37 3° ABC 8 IZ 20

Sitleofnl-^BC ( 4 6 lo)t 8885458 cOfX of decl. (i) 9.9() I53

Cof. of the lat. aSunled 9 835 1 3

Surn - 3t>.Ocooo lo>. ofthe verE f A C -verf.f AF S

57er£. f. A C-verf. f A F o.oo6868 Watural cof. of A C, the complement of the greater g 0 8I02Iz

altltudes Natural cof. of 35° I Zf z3Bo o. 8 I 7 o80

Or, _ 4 i BC + ABD-ABC b . ABD

Sine of 2 A B D ( I4 +31 to ) 2 9 40522 C.of. of the decl. of the fun 9 9915^

CCof. ot the latitude aIfilmed 9.8z25 g 3

$um 30.o0coo =log. ofthe D 8*63709

Verf. f. AD-verG. f. AF o oU67>o Natural cof. of A DX the cofsplement of the leffier g 0 730 6I

altltude, - - - * _ * ) Naturoal cof. of 35° I 2' 33", as before 0.81708 I

Thus the meri(lian diltance of the fun from tMe zenith is conzputed at 35° I28 23!t) and thls added to the filnXs declination 1 1 o i7 N. gives 46° S)' 23' for the latitude, diffcrent fironz that aXumed, and 3wearer to the true latitude; wilichX by a fecond

operatson 4

[ gl6 ]



[ 9I7 ]

operation with this latitude IlOW found, will approach llill nearer to the trutll, thus: The *um of tbe four firA logarithms in the firR ofX 19 34402

the two former operations, - -

Cofine of 46 29 23 9 83789 ABD A- ABC Slne of -X 8 42 ZS g.So6 X 3 2 AED-ABC - X _ lo 37 3°

ABC 8 4 55 ^t A B C = 4 2 2X,; | 8 84800

9 99152

9 83789 verf.f. AC-verr. n AF

IJog. 7.s2s4r

2

Verf. f. A C verf. f. A F o.oc670+

NaturaI cof. of AC, as before, O-8I02IZ Naturatcof. ot 35° 13' 21/l 0.8I69I6

The declination of the- fun added to tlaiss atnounts to 46 30' 31" the true latitude, as direftly computed, being 46° 30' I9".

HoweverX it nl t not be expeded, that this method of computation will always corlverge t1aus expe- ditiouIly to the true latitude. Had thelie obServations been made about 3 hours fooner in tlle day, and with the fame interval between themX if the firk altitude of the fun had been found I 8 55', and the fecond 33° l I', the latitude computed wcBu3.^t .ome out wider from the truth, than that aumedt orze exceeding, and the other falling ihort of t]<e trtle la titude, which will lie betvneen them, and is nearly the fame, as above. But if the leINer altittlde laad

6B 2 beea



[ 9I8 ]

Ven 29t 1 3', -and the greater 40 ; in 9 degrees of north latitude the interval between the obServations would have been nearly the fame, as before. But if this latitllde be fought by alay latitude affiumed near it, the latitude conlputed ascording to the method above, will fill more remote from the truth, than that afl*almed,, and err the fame way.

In general, the error in the latitude aXumed will bear to the error in the latitude computed, nearly tlae ratio compounded of that of the redangle under the radius and the cofine of the diRatlce from noon o-f tlae middle poillt of time between the two oltfer- vatiolls to twice fln. t ABC x fin t ABD, and tbe ratio of radv x filn. AF to fin. BC x cof. AB; inbmuch that the diRance of the middle point of ime between tlle two obfersrations from noon is to be confidered, as the limit, where tllefe com- putatlons D3ali ceaSe to cotzverge, when the reft- angle under the raclius, and the cofille of this diilance from noon {hall lre to twice fin. X A B C >; fine I A B D, (or the cofine of this diRance to the diffierenSce between tlle verSed fine of tllis diRancc from the verfed filne of half CBD) as fin. BE x cof. AB to rad. x fin. AS. And the errors in the affumed and computed latitudes fall on diffierent fides of the true latitude, when both the obServations are on the fame fide of nmn, and the seslith lies between the meridian fun and the elevated pqle; or when the ob enations are one :before, and the other after noor if the meridian fun paflês between the zenith and the eleYated pole: otherwife) they faH on the fameA l*de of the tme latitede.

HoWetJ.



-

[ 9X9 ] Howevera the dired method of computation, by

the afl lRance of the natural firles, will not be fo much more operoSe than this conlpendium, as may at firlt fssht be imagined. For the arch of a great circle being drawn through C and D, forming the triangle BCD, if the loganthmic cof1ne of BC or BD be added to the logarithmic tangent of half the angle C B D, and the logarithmic fine of B C or B D be added to the logarithmic fine of half-this angle, the firll fual is the logarithmic cotangent of the angle BCD, and the fecond the logarithmic fine of half c D, the bafe of the triangle. Then, in the triangle ^ C D, from the fidesX now all given, is to be computed the angle ACD, the difiFerence between which, and tI;le axlgle above found }3 C D, is the angle B C As when the zenith ies between the pole, and ^.e great circle through CX D; but when the zeraith lies beyond that circle, the ang3e BCA is either the fum of thok angles, or the fupplement of that falm to a circle. And, in the laR place, if twice the loga rithnwic l*1ne Qf half this angZe, and the logarithmic fines of BC and AC are added together, the film, after thrice the logarithm of the radius has been de- duEted, ss the logaritllm of h1f the excefs of tlxe mtural coflne of B C ce A C above the namral cofine- of A B, or the natural fine of the latitude, according to the trigonometrical axiom, which has been aboure referred to; for rad. x ' verf. f B-CA is equal to En. a B Cvw Alq (c).

(¢) Vide Neper. Mirif, Canon. COtlRrUdt Edsburg. s6sg,

3P' 39 The



[ 920 3 The operation for the preceding example wiII be

this: Tang. 2 C B D- 9.273225

cOf. BC 9.z91504

Sotang. B C D _ 9.5647Z9

B J: D = 87° 53 52" O , X

AD-43 S ° i\C_ 35 53 o C I3 = 20 50 5 - Sum = 99 +8 5

v Sum-49 S4 2} iSum AD- 649 z2t-

- Sum- ; Sum -

Sine22 CBD- - 9.z657X4

Sine B C 9.99 1 5 24

Sine2CD- 9.257238

z CD_ IoQ s5' 2-t-t

- 0 2320QI ? ar.co. of the fine>

, o.+4895° )

- 9 o73466} fine3- -19639038

- 9.8I95I

This half fum is the logarithnzic cofine of half A C D; therefore, half A C D is o 48° 42' 1 I", and its difference from 5 BCD will give 2 ACB 4° 4*5' 1 S"

Sine of X A C B t 8;9 I 8444 Sine of A C " 9.767999 SineofBC_ 9.991524

,oO39483-1°g 7.5964 I I

_ _

o.oo78g66 o733334s - natural cofine of B C-A C} thae is, of 4z° sotO o.72s437g r natural cof1ne of AB, or fine of- the latitude.

Therefore, the latitude is 46° 30' Ig".

But farther, as in the cafes, where the abosre ap- proximation takes place, it is an advantage, that the affiumed latitude Ahould be taken as near the true

onev



[ 92I ]

eenes as can be, the dead reckoning of the Eip need ot be intirely relied on for that purpofe; for tlle inItrumental conitrudion propoted by Mr. Collins (d) will very readily give the true latitudes as nearly as tile ilItluments uSed can exprefs: and :his method may be defcribed thus.

tRwcx Ilraight lines AB, AC B' C being dr-awn in an angle cor- 4 y rerponding with the d}Rance / jn time bctweell the obServa- z / / \5\ tions, and in oneofthe lines, 1 1 / \ as A B) tile lengths AD, A E l l / being talien fiom any fcale of 9 \ \ / 2 l tangents) ont tqual to tlle \ NGI /> taligent of half the fum and the other equal to the tangent }; ll of half tlle diffcrence of the A diIlance of tlle l-un from the pole, and firom the senith of the-firll obtervation, and alfo the pOilltS F, G, taken in the other line in the fame manner related to tlle other obServatinn; then circles being defcribed on the two dianzetcrs D E, F G, the diIlance of A from olze of the inter6(Etions of thefe circles will be the tan-- gent of Iâlf the complement of the latitude.

Moreover, as irl tlle treatife, which has given oc- cafic3n to tbis diScourCe, it is propofed fometimes to take into conI1deration the fllip's motion during-tlle time, between the ob6ervations, but imperfedly, re- gard being had to the change made by thv fhip in Jongitude only; but the change in latitude alfo may be taken into confideration in the foregoing con-

(w) ln thc plac above citet. _ _ -

Ilrudions



[ 92 ] IlruEion, by making the angle BAC equal to ttlc fum or difference of the interval of time, and the cllange in the fhip's longitllde, according as that change is made towards the weR, or tosrards tlle ea(E, and then drawing a ltne from A, not to thc interSedion of the circles; but fe, that the portions of t}wat line, terminated by eacll of the circles, may be the tangents of arches, vvhofe difference is llalf t}le cllange ill the latitude; which, if made from tlle poleX requires the portion of the line tertninated at the circles whofe diameter is DE, to be Ihorter than the other; and the contrary is required, svhe;n the motion of the lhip is tovvards tlle pole.

For determining the latitude by- calclllation, if tlzo diflance of the fun in the firft obfervation from the zenith of the fecond could be found, this cafe would be reduced to the firR, where}n the {hip is confidered as llationary. And for this purpofe, it has been pe pofed to make an additional obServatioll, by an azimuth compaSs, of the angle, the {hip's courle makes with the azimuth of the fun, when the firit aTtitude is taken; and, perhaps, the fame angle may be found with fufficient exaAnefs from the-latitude sn the firk cabServation affiumed, whether from the dead reckon- iDgx or from the foregoing confiredi0n.

Wlly I have taken no notice of the calcuIations exhibited by the author of the piece here animadverted on, as proofs of his msthod, will readily appear to thoi, who {ha}l caIl their eye upon them.

S C H O L I U M.

The axtom in trigonometry, on which the calcs lations here diScourfed of have been wewn imme-

diately



[ 923 ] diately to depend, was introduced by RegiomontanuN and is Rill retained5 as the fbundation of the prefent methods of computing logarithmically an angle from the three fides of a fpherical triangle given} though they may be dcmonIlrated more direEtly by the fUl- lowing lemma.

X

In the circle A B C} the chord A C being drawn,, and the arch A B C bifeded in B, s-f D be taken ia the circle at pleafure, and tl lines B E, D F drawn alSo at pleafure parallel to each otller, one terminated by the circle in E, and the otller by the chord AC in F, then AD5 DC being drawn, BE x DS is AD x DC.

Draw D fI perpendicular to A Cs and alfo the diameter B G, joining E G Then the angles FDH E B G are equal, and the angles iC) HF B E G righe Therefore BE is to BG as DH to DEi;, and BE x DF

B G uc D H A D x D C

vOL LIo 6 c NOws

- w w J



[ 924 ] SI^! es StOW, isl tha i

c, < fphet-ical triangle

A />z / \ B, and to the / t ,/ zj / 19 poles A, C, let / / / / / \ the iSemicirc]Cr

/ /// EBF, GBDbe // >, - deScribed, and

//// tile alch A C gL/ / conzpleted to a \ / cilcle. Then the

X / femicirclesEBE, - \ : / GBD will be

both perpendi cular to tlle plalle of this circle, cut it iIl thtiE

diameters E-F, GD, and their common interfeEtion BS be alSo perpendicular to it, and conSequerltly perpendi-Cvilar to G D, the diameter of tlle Setnicircle G B D, svhereby G B, DB Xbeing Joined, G D x G H is equal to GBq, and GD x DH equal to BDq. Agairl, A K being drawn parallel to -G D, the arcIl A C Il is double A C, G C D double C B, E A I) tlue futn of all tlle three f1des of the triangle, E G -_ 15:D DG (CB),GF=AF (AE)+CG (CD)-AC _E1:)-2 AC, and FD = ED-EF (2 AB).

Then E G, G F, F D, E D being joined; in the firll place, AK is to t)G as AK x GH to 13G x GIS, alfo AI -to D G as AK X DG to D G 9) bllt AK x GH

EG x G F by tlle preceding lemma, and DS x GH =GBq; wherebyAKxDG istoDGq asEtXxGF to G B s. Hence A K being tw;ce the fille of tlae al Ch

AC, DG twice the fine of the arch GC) equal tc B C} 1S G twice the flne of half tlle arch E A G, G F

twice



[ 925 ] twice the f1ne of half the arch G F, azd G B twice the I^}ne of 11alf the angle A C B to tlle radius of the circle G B D, which is half G D ; fin. A C x fin. B C is to rad.q as fin. 2 ED-B(: x fin. I ED-AC to fin. t A (: Blq. And this is Napeir's firlt methczd of finding an angle firom the three fides given (e), as it is ufually delivered.

Again, AK ls to DG, or AK x DG to DGs, as A 1( x D H (E D x D F) to D G x D H, or D E q. Therefore, D B being twice the fine of half the angle B C D, or twice the cof1ne of half the angle A C B to the radius of the circle G B D, as f1n. AC x fin. C B 4 to rad.q fo is 2 F, D x fin. 2 E D-A b to cof. t AC Blq. And this is Napeir's fecond method (f).

And laRly, the cofine of an arch or angle being to its fine, as radius to the tangent, f1n t E D

x fln T ED AS will be to fln.- 12 ED-BC

x f1n, T E D -- - AC as rad9 to tang. t A C Bl§, which is a third method added by Gellibrand (g).

Moreover, In plain triangles, from the three fides given may

an angle be found by a proceEs f1milar to each of theSe, as follows:

(e) Mirific. Canon. Log. defcrip. lib. ii. c. 6. § 8.

( g) Trigon. Britan. p. 75.

6C 2 In



Tn the plain triangle ABC, if on the centers A and C be deScribed circles through B, cuttirlg A C in E, , G, and D, the perpendicular B H beillg let fall on ACf EHxHF is equa1 to GHxHD. Tllere_- fore H F: G H:: H D: H , and by compof1tioz GS: GH :: ED: HE, asld: :ED-GE : EGy alfoHF:GF; :HD:ED, and ::DE:ED GF.

Moreover ELD i$ = AB + AC + CB, 1SG = ED-2CB, FD ED_ 2AB, ED-GF =:EG+ ED - 2ED-2CB--"- 2AB zACb tIence CF is to GH as 2AC to ES, whence GF x EG is = 2AC xGH; and DH is to EL) as D i' to Z A C, that isX 2 A C x D IS will be E D X D Fe

Thus 2AC: DS (:: aAC x DG: DGq)- *: 2AC x GEI (EG x CF) : DGH (Gl3S) ancL AC x CB : rad.q ;: , ED -- CB x 5 ED --- AC- t {in. 2 A (: sg q

L;kesvife AC: DG ( :: 2;AC x DG ::DGs)

:: ACx DII (:EZ Dx DF) : GDH (DBq); therefore^

ACxC:B: rad.q;: t-EDxflED-AB:cof.- AC8lq And in the laft place 9 E D x 12 E I) - A B

: - E D-CB xTED_AC; ; radq; tang. -ACE%9 5 Thze

[ 926 ]



[ /927 ] Tlais laIl propefition was filR delivered by G¢lit

lbrand, and the other tsnro not lolzg afiter by Oughtred in their refpeEt£ve treatifes of Trigonoxnetrye

BuT it tnay farther here l3e llot£d, that fiom tBro fide.s of a plain triangle, as A C, C B} mJith tne ang;le between them ACB) the thlrd fide AB may be found either by the firit or fecond of the :foregoing proportions > for by the firll may be foulld E G x G F that is, A Fq-A G9, and by the fecond E D x D F or A Dq-A Fq; whence by a table ef fquares A F (= AB) may be readily fbund-> for AG is eclual t

AC .11 B{S, and AD AC + CB.

In like manner in the fphe_ C rical triangle firotn the Iides AC, CB,> arld the angle ACB, / \ tnay the third Iide be exped- / \ tioully fiound with tlae aid of a / \ table of natural f1nes, by the /

axiom geferred to in the be- < i3)}

ginning of this difcourfe) as tnociclled for tlwis p poSe by lD{apeir, in t}le propof1tionX wIzich has bce already referred to (Z3}, vlaereby the aXiotn is le duced to this analogy, Rad.q . rl. A C x fin. C B :; lill. w A C &

verf.f. AB-verf.f. AC: z CI3 *rad.x . ( )

NIAY

(:^) Mirif. Canon. Con{tr. p. 9. (i) An example of this method has been given above in the tr;-

angle 1t B C in p. gzo And itR default of a table of natural fines) £he bafe A B might have been found t}}us. The futn of twce thc 3egarsthmic {;ne of -g; A C Ba addcd to the logarithmic fings of A 6



[ 828 ] SIAY it lzere be far>her added, that thus all the

cafes of oblique Epherical tria.ngles nzay be Colved witllout dividing them into reEtangular ? JFor when two fldes !nd the angle illcludcd) or wlaen two angles and tlae Elde betvVeerl thWm are given, the two other angles or fidt>S tBay be found by the proportions of Napeir and I31igggs, etl ereby the ot3acr tw> angles or fIdes are found together (k); the proportions, when two fiIdvs with the ilIcluding angle are given, being thefe, C) < AC + CB cof Ac cn CB ;; cotang ̂ ACS

2 2 CAB + CBA 0

* tan¢ ^, * - a Arld

Sin A C + C B fIn. A C z C B;, cotangX ^ A C B

CA13 cnCBA : tang. 2 '

The fecond of theSe has been delnonRrated with great concifenefs by Dr. Halley, from the principles of the Rereographic projedion of the fphere (/); and the firA is derived by hitn from the fecond, but not

__

and B C amounted to 37.59641I. The logarithmic Eme of

A C cn B C (9 562+68) deduEted from half this fium (I8.7982I I )

leaves 9.z35737. This number fought among the logarithmic tangents exhibits for its correfpondent fine g.22g4co, which being deduflced from the forefaid half fum (IS 792II) leaves 9.568805 which is the logarithmic fine of ZIe 44 so", equal to half AB.

(k) llide Neper. Mirif (Canon. Confir, pag. penult. (I) See Jones's SynopEisl p. 281.

Sal to geth e r



[ 929 ]

altogetlaer with the fame fuccefs, by his not obServing that if the bafe and one of tlle fides be completed t femicircles, t}e 1êcond Qt' the two proportions irl tilc fupplemental triarlgle therjce formed leads diredily tc the fisR in tlle origil.al triangle.

And to conclude, if two fidcs, and tbe angle op pcohlte to one of them) or two allgles srith the fi(Sc oppoilte to orwe, were given; wllen the other oppotite part is found fronl the proportion between the fincs of parts oppofite, the remaining angle) or fide, may be found by eitller of the two p-roportions foregoi-X3g

LXXXIf. Sn Sccount cf rhe Plants -Halctia anal Gardenia : In a Leter fkoan John Ellis, t/qf; F. R. 3. to I'lailip Carteret Webb, &f; F.;R.Sv

Dear Sir, eadNov2oRirou nruCt have oltfcrvee3) that as

6O sL the fpirit:of plantilzg; has increaScdf n this kilagtRoln3 the fludy of bolany has become^ nzore fiffiioIlable; the xlYorks of tlae celebrated Lin- laus) heretofUre looked on as capricious and firangeX are IlOW in vlle hatwds of evcry man, who wies to ftldv tlae crder of nature.

Tlae gleat variety of plantsJ which you have in- troelllced lnto your gardela from North America, as wc11 as fronz many other parts of the world muR: ;ive you doul)le pleafllre, when you wiew them ranged Xx proper ordtrX a1ld 3udicioufiy namedO

[ 929 ]

altogetlaer with the fame fuccefs, by his not obServing that if the bafe and one of tlle fides be completed t femicircles, t}e 1êcond Qt' the two proportions irl tilc fupplemental triarlgle therjce formed leads diredily tc the fisR in tlle origil.al triangle.

And to conclude, if two fidcs, and tbe angle op pcohlte to one of them) or two allgles srith the fi(Sc oppoilte to orwe, were given; wllen the other oppotite part is found fronl the proportion between the fincs of parts oppofite, the remaining angle) or fide, may be found by eitller of the two p-roportions foregoi-X3g

LXXXIf. Sn Sccount cf rhe Plants -Halctia anal Gardenia : In a Leter fkoan John Ellis, t/qf; F. R. 3. to I'lailip Carteret Webb, &f; F.;R.Sv

Dear Sir, eadNov2oRirou nruCt have oltfcrvee3) that as

6O sL the fpirit:of plantilzg; has increaScdf n this kilagtRoln3 the fludy of bolany has become^ nzore fiffiioIlable; the xlYorks of tlae celebrated Lin- laus) heretofUre looked on as capricious and firangeX are IlOW in vlle hatwds of evcry man, who wies to ftldv tlae crder of nature.

Tlae gleat variety of plantsJ which you have in- troelllced lnto your gardela from North America, as wc11 as fronz many other parts of the world muR: ;ive you doul)le pleafllre, when you wiew them ranged Xx proper ordtrX a1ld 3udicioufiy namedO



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