Spherical Trigonometry Reduced to Plane, by Francis Blake, Esq. F. R. S.Author(s): Francis BlakeSource: Philosophical Transactions (1683-1775), Vol. 47 (1751 - 1752), pp. 441-444Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/105087 .

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[ 44t ] LXXIV. Spherical Trigonometry redaced to

Plane, by FrancistBlake, &2' F. R. S. Read May 7 tT is obServable, that tlle qnalogies of

I752* 1 fplaerical trigonometry, exclufisre of the terms co-f1ne and co-tangent, are applicable to planeX by only changing the expreilon, filne or tallgent of fide, into the flngle woxd, flde *: fo that the buf1Ilefs of plane trigorlotnetry, like a corollary to the other, is thence to be inferr'd. And the reafon of this is obvious; for analogies raired not only from the con- fideration of a triallgular figure, but the cllrvature alfo, are of confequence more general; and thoS the latter Ihould be held evanei:cent by a diminution of the furface, yet wlaat depends upon the triangle, will neverthelefs retntin. Thefe things may haxre been obServed, I fay ; but upon revifing the fub3e&, it filr_ ther occurr'd to me, and I take it to be new, that from the axioms of only plane trig<3nometryn and almoll independent of folids, and the dodrine of the fphere, the Epherical cafes are likewife to - be folved.

Suppofe, firll, that the three fides of a fpherical triangle, a b d (Fig I.) are given to find an angle, s; which caSe will lay open the method, and lead on to the other cafes, in a way, that to me appears the mo* natural. It is allow'd, that the tangents, ve, a f, ofthe fldes, a d, a b, including m aIlgle, at make a plane angle equal to it; and it is evident) that the other fildeX db, determines the angle made 6y tlae fecants ce, cf, at c the centre of the rphere; whence the dikance, ef, betwixt thz tops of thofe fecants, is

K k k given i s.

* See M. De la Cailless remark at the end of the fpherical trigono- metry prefix'd to his Elements of ARronomy

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[ 442 ] given by cafe the fifth of oblique plane triangles (lSee: BeynesXs Totgoxom.) ̂rhich with the a-fol-efaid tan- gents, reduces it to caSe the 6tll of obliqut plane tri- angles alSo * : and thus tllis X Ith cafe of oblique tli- angles, fo intricate hitherto, becomes perfeftly cafy. The I 2th cafe is reducible to the I I th, and the re{t, whether right-angled, or olDlique, we are authorifed to look upon as reducible to right-angled trianglesZ wllofe f1des are not quadrants, but either greater or lefs than fuch. -Conceive therefore, now, in a right- angled i:pherical triangle, gAh (Fig. 2.) that the tan- gent, gm, and fecant, em, of elther leg, gk, iS al- ready drawn; and in the points aw, oftheir ulliona drasv a perpendicular, ml) to ew, the Secanta di- redly above the other leg, eDiZ. a perpendicular ta the plane of the fecant arld tangent, that it may be perpendicallar to both (Eucl. +, I I ) ; for then will the tangent, gl, of the hypother:uSea gh, dravwrn from the fame point, whIch that of the leg wasX conIRantlv terminate in the perpendicular line, that the radius and tangent may make a right-angle (Etlcl. I8, 3). Whence thefe tangents, g > g l, and the perpendicular line, ml, together with the fecants,, c , c 1, will evidently form two right-angled plane triangles, g m 1, c xn /; and to one or other of thefe the ipherical cafes are eaflly transfert'd. Thus} if in the fpherscal triangIe} gk> the hypothenureX gh,, bafe, gk, and angle, g, at the baSe, be the parts given-and required, when any two ase given, the

thlrd

* The angle to be found in this cafe mi alvfays be that formed byie two tangents.

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t -

l 443 third may be determined by means of a plane trianZ gle; and at a flngSe operation. We have, {QR illu Itance in the right-angled p1an$ triangle, g azl brmed as abore, the bafe, g-ma and hypothentre, glZ to find, by cafe the sth of nght^sngled plan: triangles} the angle included, whtch Is tShe fame at on the fphere. And then xf the bafe, g k, the ang4e} g, at tlle baSes and perpendicular kb, be the fpheri- cal parts given and requirel, or if the angtes, g and b and the hypothenuiE? gb be the parts gwen and required, we have only that former proportion of the hypothenuSe and bleX and gnglezat the hafe, in the tria2lgles, PND, DFG, obtaine.d by the compleme:ss to transfer to. the e. I3ut fer condly, fuppofe the lkblerical propttiNi is of tht three fides, any two betng given, the therd may be alSo found at a linglv operation, in the Sed right- angled plane triangle, c m 1 form'd as abow*. Wc have} fw inRanceZ the hypothenuCe and baSe, tt cwt sz. the fecant of the fpherical hypothenuSe and baSe g higk, to find, by the sth of rightaed plane trian- gles, the angle, c, at the center-, ob is the fure of k h, the f1de that was fought. And then again if the hypothenufie, one leg, and the oppoF fite angle be the fpherical parts given and required; or if the two angles and a leg be the parts given and required, we have only the former proportion of the three fides in the triangles, PPXD, DFGX obtained bythecomplements, tstransfer to the plane. Whencet the fix proportions of right-angled Epherical triangles being comprehended in this method, it is fully demon firaul at all the cafes of theSe triangles are fo to !3e ref;blved.

Kkk 2 Tn

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[ 444 ] The fame might be deduced without the method

of complements, but neither in fo lSort nor fatisfac- tory a way, and it Ihall therefore be omitted. I have communicated this upon account of its perfpicuity, and fuppofing, that in an age fo greatly advanced in mathematical learniIlg, the leaIl hint af what is new would not be-unacceptable.

Figo I. Fig. 2. fig. 3Xm

ueens Square, Ws, May 7 7S*

^;xxv

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