vol. XV.] .;! PHILOSOPHICAL TRANSACTIONS. 227 



we take the intermediates; fori, 14., 2^, 3-^, &c. min. or the doubles of these, 

 1, 3, 5, 7, &c. min. Which yet, because on the convex side of the curve, 

 would be somewhat too little. But any of these ways are exact enough for the 

 use intended, as causing no sensible difference in the chart. If we would be 

 more exact, Mr. Oughtred directs, and so had Mr. Wright done before him, 

 to divide the arch into parts yet smaller than minutes, and calculate secants 

 adapted to them. Since the arithmetic of infinites has been introduced, and, 

 in pursuance thereof the doctrine of infinite series (for such cases as would 

 not, without them, come to a determinate proportion), methods have been 

 found for squaring some such figures, and in particular the exterior hyperbola, 

 by way of continual approach, by the help of an infinite series. As, in the 

 Philos. Transact. N" 38,* and my book De Motu, cap. 5. prop, 31. In 

 imitation of which, it has been desired by some, that a like quadrature for this 

 figure of secants, by a proper infinite series, might be found. 



In order to which, put r for the radius of a circle, s for the right sine of an 

 arch or angle, v for the versed sine, c = r — v = ^ : r'^ — j^ : for the co-sine 

 or sine of the complement; r — u = a- for the secant; t for the tangent. Then 

 is c : r :: r : 0-, or a- = — the secant ; and c: s :i r: t, or f := — the tan- 

 gent. Now, if we suppose the radius C P, fig. 7, divided into equal parts, and 

 each of them = -f r ; and on these to be erected the co-sines of latitude LA: 

 then are the sines of latitude in arithmetical progression ; and the secants answer- 

 ing to them, LS= — , But these secants, answering to right sines in arith- 

 metical progression, are not those that stand at equal distances on the qua- 

 drantal arch extended, as fig. 6, but standing at unequal distances on the same 

 extended arch; viz. on tho«e points thereof, whose right sines, whilst it was a 

 curve, are in arithmetical progression, as fig. 8. 



To find therefore the magnitude of R E L S, fig. 6, which is the same with 

 that of fig. 8, supposing E L of the same length in both, however the number 

 of secants therein may be unequal ; we are to consider the secants, though at 

 unequal distances, fig. 8, to be the same with those at equal distances in fig. 7, 

 answering to sines in arithmetical progression. Now these intervals, or por- 

 tions of the base, in fig. 8, are the same with the intercepted arches, or portions 

 of the arch, in fig. 7; for this base is only that arch extended. And these 

 arches, in parts infinitely small, are to be reputed equivalent to the portions of 

 their respective tangents, intercepted between the same ordinates; as in fig. 7j 

 9; that is, equivalent to the portions of the tangents of latitude: and these 



• Vol, 1. p. 273, of thi* abridgment 

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