for the resolution of Plaine and Sphericall Triangles.

After twelve lines of further remarks on this point he adds:

Hence from the forme, I have called it a Ring, and Grammelogia by annoligie of a Lineary speech; which Ring, if it were projected in the convex unto two yards Diameter, or thereabouts, and the line Decupled, it would worke Trigonometrie unto seconds, and give proportionall numbers unto six places only by an ocular inspection, which would compendiate Astronomicall calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability to health and time.

The unnumbered page following page 22 contains the patent and copyright on the instrument and book:

Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs an Instrument called Grammelogia, or The Mathematicall Ring, together with a Booke so intituled, expressing the use thereof, being his owne Invention; we of our Gracious and Princely favour have granted unto the said Richard Delamain and his Assignes, Privilege, Licence, and Authority, for the sole Making, Printing and Selling of the said Instrument and Booke: straightly forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument or Booke within any our Dominions, during the space of ten yeares next ensuing the date hereof, upon paine of Our high displeasure. Given under our hand and Signet at our Palace of Westminster, the fourth day of January, in the sixth yeare of our Raigne.

Delamain’s later designs, and directions for using his instruments

In the Appendix of Grammelogia III, on page 52 is given a description of an instrument promised near the end of Grammelogia I:

That which I have formerly delivered hath been onely upon one of the Circles of my Ring, simply concerning Arithmeticall Proportions, I will by way of Conclusion touch upon some uses of the Circles, of Logarithmall Sines, and Tangents, which are placed on the edge of both the moveable and fixed Circles of the Ring in respect of Geometricall Proportions, but first of the description of these Circles.

First, upon the side that the Circle of Numbers is one, are graduated on the edge of the moveable, and also on the edge of the fixed the Logarithmall Sines, for if you bring 1. in the moveable amongst the Numbers to 1. in the fixed, you may on the other edge of the moveable and fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto 6.6. and each degree subdivided, and then over the former divisions and figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5. 4. 3. 2. 1. each of those divided by small points.

Secondly, (if the Ring is great) neere the outward edge of this side of the fixed against the Numbers, are the usuall divisions of a Circle, and the points of the Compasse: serving for observation in Astronomy, or Geometry, and the sights belonging to those divisions, may be placed on the moveable Circle.

Thirdly, opposite to those Sines on the other side are the Logarithmall Tangents, noted alike both in the moveable and fixed thus 6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or divisions serve also for their Complements to 90. so 40 gr. stands for 50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both in the moveable and fixed is also divided into parts. As for the degrees which are under 6. viz. 5.4.3.2.1. they are noted with small figures over this divided Circle from 45.40.35.30.25. &c. and each of those degrees divided into parts by small points both in the moveable and fixed.

Fourthly, on the other edge of the moveable on the same side is another graduation of Tangents, like that formerly described. And opposite unto it, in the fixed is a Graduation of Logarithmall sines in every thing answerable to the first descrition of Sines on the other side.

Fifthly, on the edge of the Ring is graduated a parte of the Æquator, numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree both of the Æquator, and Meridian are subdivided into parts; these two graduated Circles serve to resolve such Questions which concerne Latitude, Longitude, Rumb, and Distance, in Nauticall operations.

Sixthly, to the concave of the Ring may be added a Circle to be elevated or depressed for any Latitude, representing the Æquator, and so divided into houres and parts with an Axis, to shew both the houre, and Azimuth, and within this Circle may be hanged a Box, and Needle with a Socket for a staffe to slide into it, and this accommodated with scrue pines to fasten it to the Ring and staffe, or to take it off at pleasure.

The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and V make no reference to the dispute with Oughtred and may, therefore, be assumed to have been published before the appearance of Oughtred’s Circles of Proportion. On page 53, “To the Reader,” he says:

. . . you may make use of the Projection of the Circles of the Ring upon a Plaine, having the feet of a paire of compasses (but so that they be flat) to move on the Center of that Plaine, and those feet to open and shut as a paire of Compasses . . . now if the feet bee opened to any two termes or numbers in that Projection, then may you move the first foot to the third number, and the other foot shall give the Answer; . . . it hath pleased some to make use of this way. But in this there is a double labour in respect to that of the Ring, the one in fitting those feet unto the numbers assigned, and the other by moving them about, in which a man can hardly accommodate the Instrument with one hand, and expresse the Proportionals in writing with the other. By the Ring you need not but bring one number to another, and right against any other number is the Answer without any such motion. . . . upon that [the Ring] I write, shewing some uses of those Circles amongst themselves, and conjoyned with others . . . in Astronomy, Horolographie, in plaine Triangles applyed to Dimensions, Navigation, Fortification, etc. . . . But before I come to Construction, I have thought it convenient by way introduction, to examine the truth of the graduation of those Circles . . .

These are the words of a practical man, interested in the mechanical development of his instrument. He considers not only questions of convenience but also of accuracy. The instrument has, or may have now, also lines of sines and tangents. To test the accuracy of the circles of Numbers, “bring any number in the moveable to halfe of that number in the fixed: so any number or part in the fixed shall give his double in the moveable, and so may you trie of the thirds, fourths &c. of numbers, vel contra,” (p. 54). On page 55 are given two small drawings, labelled, “A Type of the Ringe and Scheme of this Logarithmicall projection, the use followeth. These Instruments are made in Silver or Brasse by John Allen