After describing a modification of the above arrangement, he adds, “many other formes might be deliverd, about this single projection” (p. (32)). Proceeding to the “enlarging” of the circles in the Ring, to, say, the “Quadruple to that which is single, that is, foure times greater,” the “equall parts” are distributed over four circles instead of only one circle, but the general method of graduation is the same as before (p. (33)); there being now four circles carrying the logarithms of numbers, and so on. Next he points out “severall wayes how the Circles of the Mathematicall Ring (being inlarged) may be accommodated for practicall use:” (1) The Circles are all fixed in a plain and movable flat compasses (or better, a movable semicircle) are used for fixing any two positions; (2) There is a “double projection” of each logarithmic line “inlarged on a Plaine,” one fixed, the other movable, as shown in his first figure on the title-page, a single index only being used; (3) use of “my great Cylinder which I have long proposed (in which all the Circles are of equall greatnesse,) and it may be made of any magnitude or capacity, but for a study (hee that will be at the charge) it may be of a yard diameter and of such an indifferent length that it may containe 100 or more Circles fixed parallel one to the other on the Cylinder, having a space betweene each of them, so that there may bee as many mooveable Circles, as there are fixed ones, and these of the mooveable linked, or fastened together, so that they may all moove together by the fixed ones in these spaces, whose edges both of the fixed, and mooveable being graduated by helpe of a single Index will shew the proportionalls by opposition in this double Projection, or by a double Index in a single Projection” (p. (36)). Next follows the detailed description of his Ring “on a Plaine, according to the diagramme that was given the King (for a view of that projection) and afterwards the Ring it selve.” The diagram is the large one which we mentioned as inserted between pages (23) and (24). The instrument has two circles, one moveable, upon each of which are described 13 distinct circular graduations. The lines on the fixed circle are: “The Circle of degrees and calendar,” E. “Circle of equall parts, and part of the Equator, and Meridian,” TT. “The Circle of Tangents,” S. “The Circle of Sines,” D. “The Circle of Decimals,” N. “The Circle of Numbers.” The lines on the movable circle are: N. “The Circle of Numbers,” E. “The Circle of equated figures, and bodies,” S. “The Circle of Sines,” TT. “The Circle of Tangents,” Y. “The Circle of time, yeares, and monethes.” On pages (84)-(88) Delamain explains an enlargement of his Ring for computations involving the sines of angles near to 90°. On page (86) he says: