After elaborating this last point and explaining the decimal subdivisions on the scales of the movable circle, he says that “the numbers and divisions on the fixed Circle, are the very same that the moveable are, . .” There is no drawing of the slide rule in this publication. The twenty-two numbered pages give explanations of the various uses to which the instrument can be put: “How to performe the Golden Rule” (pp. 1-3), “Further uses of the Golden Rule” (pp. 4-6), “Notions or Principles touching the disposing or ordering of the Numbers in the Golden Rule in their true places upon the Grammelogia” (pp. 7-11), “How to divide one number by another” (pp. 12, 13), “to multiply one Number by another” (pp. 14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21), “How to performe the Golden Rule” (the rule of proportion) is explained thus:
Seeke the first number in the moveable, and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed.
If the Interest of 100. li. be 8. li. in the yeare, what is the Interest of 65. li. for the same time.
Bring 100. in the moveable to 8. in the fixed, so right against 65. in the moveable is 5.2. in the fixed, and so much is the Interest of 65. li. for the yeare at 8. li. for 100. li. per annum.
The Instrument not removed, you may at one instant right against any summe of money in the moveable, see the Interest thereof in the fixed: the reason of this is from the Definition of Logarithmes.
These are the earliest known printed instructions on the use of a slide rule. It will be noticed that the description of the instrument at the opening makes no references to logarithmic lines for the trigonometric functions; only the line of numbers is given. Yet the title-page promised the “resolution of Plaine and Sphericall Triangles.” Page 22 throws light upon this matter:
If there be composed three Circles of equal thicknesse, A.B.C. so that the inner edge of D [should be B] and the outward edge of A bee answerably graduated with Logarithmall signes [sines], and the outward edge of B and the inner edge of A with Logarithmes; and then on the backside be graduated the Logarithmall Tangents, and againe the Logarithmall signes oppositly to the former graduations, it shall be fitted 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: