( * 6 ? )

^5- To find therefore cke trtie Magnitude os the Parallelograms ( or Seg-ments of the Figure; ) we must either Protract the Equal Segments os thcBase. . Fig. no. {itt such Proportion as is the respecti ve Tangents to theSine) to make them Equal tothose of Fig. zu.

3 6 . Or eise ( whieh is äquivalent ) retaining the Equal Intervals of Figiiio. Protract the Secants in the fame Proportion. (For either way, thelnter-cepted Rectangles or Parallelograms will be equally Increased) as LM. Fig. z iz'

37. Namely; as the Sine ( of Latitude ) to its Tangent; se is the Se-cant, to a Fourth ; whieh is to stand ( on the Radius equally divided ) in-stead of that Secant. <

(::S:R.)

R 2 R 3

2 V“ = R 2 — S* LM ’

3 8. Whieh therefore are as the Ordinates in ( what I call Arith. Infin,Prop. 104. ) Rpciproca Secundanorum : Supposing 2 1 to be Squares in the Or-der of Secundanes.

of = R 1 — S 2 ; and the Sines S, in

Arith-

39. This ( because ^ _ _ , _ .... _ .

metical Progression ) is Reduced ( by Division ) into this Infinite Serie s

n S 2 S4 s 6 ,

R.-f- -j--1- —, fßc.

R R 3 R y’

40. That is, ( putting R — 1.) 1 -i- S 2 -I- S* •+ S^, c£e.

41. Then ( according to the Arithmeticis of Infinites ) we are to InterpretS,: successively, by 1 8, iS-, 3S ,F3c. till we come to 8, thegreatest. Whiehtherefore Represents theNumberof all.

41. And because the sirst Member doth Representa Series of Equals, theSecond of Secundans j the Thitd of Quartans, Cc. Therefore the First Mem-ber is to be Multiplied, by S j the Second by 3 S i the Third by ] S , theFourth by 7 8; Gfc.

43. Whieh makes the Aggregate, S -+■ 3' S? -+■ , S* 4 -\ S 7 -+• l S^, c£r.— ECLM.

44. This C because S is always lese than R ■= 1 ) may be so far conti-nuedj.till some Power of S become so finali as that it .( and ali whieh follov/it ) may be fafely neglected.

45*« Now (to sit this to the Sea-Chart , according to Mr. iVrights Design)having the proposed Paralie} (of Latitude ) given ; we are to find (by the-Tr»-gonometrical Canon) the Sine of soch Latitude j and take Equal to it, CL — 8.And, by this find the Magnitude of ECLM, Fig. ziz. that is of - REL f. .*Fig. in. that is, of REL/. Fig. 109. And then, as RRLE ( or so manytimes the Radius) to REL^ ( the Aggregate of all the Secants ; ) so must 'be a like.Arch of thc iEquator ( Equal to the Latitude proposed,) to theDistance^of soch Parallel, (representing i\\e.Latitude in the Chart) fromthe./Equator: > Whieh is the thing required.

46. The fame may be obtained, in like manner, by taking the Vei sed Sinesin Arithmetical Progression. For if the Rigbt Sines ( as here ) beginning atth« ^Equator, be in ArithmeticaL Progression, as, 1, z, 3, 0c. Then will

th« ^Equator, be in ArithmeticaL Progression, as, 1, z, 3, tUc. Then will