S18 PHILOSOPHICAL TRAKSACTIONS. [aNNO 1685. 



portions of tangents, are to the equal intervals in the base, as the tangent of 

 latitude is to its sine. 



To find therefore the true magnitude of the parallelograms, or segments of 

 the figure, we must either protract the equal segments of the base, fig. 7, in 

 such proportion as is the respective tangent to the sine, to make them equal 

 to those of fig. 8; or else, which is equivalent, retaining the equal intervals of 

 fig. 7, protract the secants in the same proportion : for, either way, the inter- 

 cepted rectangles or parallelograms will be equally increased: as LM, fig. 9 ; 

 viz. as the sine of latitude is to its tangent, so is the secant, to a fourth ; 

 which is to stand on the radius equally divided, instead of that secant : that is, 



as* : — ore: r :: — : -r — - = LM, fig. 0. 



And this, because c^ := r^ — s^, and the sines s in arithmetical progression, 



J* /** s^ 

 is reduced by division! into this infinite series, r -\ 1 — j -| — j-, &c. That 



is, putting r = 1, the series 1 -^ s"^ + s* + s^ +, &c. 



Then, according to the arithmetic of infinites, we are to interpret s, suc- 

 cessively, by 15, 2s, 3s, &c. till we come to s, the greatest, which therefore 

 represents the number of all. And because the first member represents a series 

 of equals ; the second, of secundans ; the third, of quartans, &c : therefore 

 the first member is to be multiplied by s ; the second by -^s ; the third by -^ ; 

 the fourth by -f j, &c: which makes the aggregate, * + i-** + i** -j- i«' + 

 j^s^ -\- &c. = E CLM, fig. 9. And this, because * is always less than r or 1, 

 may be so far continued, till some power of s become so small, as that it, and 

 all which follow it, may be safely neglected. 



Now, to adapt this to the sea-chart, according to Mr. Wright's design : 

 having the proposed parallel of latitude given ; we are to find, by the triogono- 

 metrical canon, the sine of such latitude, and equal to it, take C L = j ; and 

 by this find the magnitude of E C L M, fig. 9 ; that is, of R E L S, fig. 8, or 

 of R E L S, fig. 6, And then, as R R L E, or so many times the radius, is 

 to R E L S, the aggregate of all the secants ; so must a like arch of the equator 

 equal to the latitude proposed, be to the distance of such parallel, representing 

 the latitude in the chart, from the equator : which is the thing required. 



The same may be obtained, in like manner, by taking the versed sines in 

 arithmetical progression. For if the right sines, as here, beginning at the 

 equator, be in arithmetical progression, as 1,2, 3, &c.; then will the versed 

 sines, beginning at the pole, as being their complements to the radius, be so 

 likewise. 



The Collection of Tangents. — ^The same may in like manner be applied to the 

 aggregate of Tangents, answering to the arch divided into equal parts. For, 



