VOL. XV.] PHILOSOPHICAL TRANSACTIONS. 22* 



General Preparation. — Because curves are not so easily managed as straight 

 lines ; the ancients, when they contemplated figures terminated, at least on one 

 side, by a curve line, either convex or concave, as AFKE, fig. 1, i, pi. 7, 

 they often made use of some such expedient as this following, viz. By parallel 

 straight lines, asAF, BG, C H, &c. at equal or unequal distances, as there 

 was occasion, they divided it into as many segments as they thought fit; or 

 supposed it to be so divided : which segments were less by one than the number 

 of those parallels. To each of these parallels, wanting one, they fitted paralle- 

 lograms, of breadths equal to the intervals, between them and the next follow- 

 ing: which thus formed an adscribed figure made up of those parallelograms. 

 Then, if they began with the greatest, and therefore neglected the least, such 

 figure w_as circumscribed, as fig. 1, and therefore larger than the curvilinear 

 figure proposed. If with the least, neglecting the greatest, the figure was in- 

 scribed, as fig. 2, and therefore less than that proposed. But, as the number 

 of segments was increased, and thereby their breadths diminished; the differ- 

 ence of the circumscribed from the inscribed figure, and therefore of either 

 from that proposed, continually decreased, so as at last to be less than any 

 assigned. On which they grounded their method of exhaustions. In cases 

 wherein the breadth of the parallelograms, or intervals of the parallels, is not 

 to be considered, but their length only; or, which is much the same, where 

 the intervals are all the same, and each reputed = 1 : Archimedes, instead of 

 inscribed and circumscribed figures, used to say. All except the greatest, and 

 All except the least. As Prop. 11. Lin. Spiral. 



The particular Case. — Though it be well known, that in the terrestrial globe 

 all the meridians meet at the pole, as E P, E P, fig. 3, whereby the parallels to 

 the equator, as they be nearer to the pole, continually decrease: and hence a 

 degree of longitude in such parallels is less than a degree of longitude in the 

 equator, or than a,degree of latitude; and that, in such proportion, as is the 

 cosine of latitude, which is the semidiameter of such parallel, to the radius ot 

 the globe, or of the equator: yet it has been thought fit, for certain reasons, to 

 represent these meridians in the sea-chart, by parallel straight lines; as Ep, Ep. 

 Whereby each parallel to the equator, as LA, was represented in the sea-chart, 

 by la, as equal to the equator EE ; and a degree of longitude therein, as large 

 as in the equator. By this means, each degree of longitude in such parallels, 

 was increased beyond its just proportion, at such rate as the equator, or its 

 radius, is greater than such parallel, or its radius. But in the old sea-charts, 

 the degrees of latitude were still represented, as they are in themselves, equal 

 to each other, and to those of the equator. By which, among many other 

 inconveniences, as Mr. Wright observes, in his correction of errors in naviga- 



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