and the hair shall cut on the equal parts 81 16; and the contrary, if you have a decimal fraction, and would reduce it to a proper fraction, the like may you do for shillings, and pence, and pounds, and ounces.

The uses of the lines follow.

As to the use of these lines, I shall in this place say but little, and that for two reasons. First, because this instrument is so contrived, that the use is sooner learned then any other, I speak as to the manner, and way of using it, because by means of first second and third radiusses, in sines and Tangents, the work is always right on, one way or other, according to the Canon whatsoever it be, in any book that treats of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in Oughtred from page 64 to 107.

Secondly, and more especially, because the more accurate, and large handling thereof is more then promised, if not already performed by more abler pens, and a large manuscript thereof by my Sires meanes, provided many years ago, though to this day not extant in print; so for his sake I claiming my interest therein, make bold to present you with these few lines, in order to the use of them: And first note,

1. Which soever of the two legs is set to the first term in the question, that I call the first leg always, and the other being set to the second term, I call the second leg . . .

The exact nature of the contrivance with the “two legs” is not described, but it was probably a flat pair of compasses, attached to the metallic surface on which the serpentine line was drawn. In that case the instrument was a slide rule, rather than a form of Gunter’s line. In his publication of 1661, as also in later publications,[13] John Brown devoted more space to Gunter’s scales, requiring the use of a separate pair of compasses, than to slide rules.

Changes introduced by William Leybourn

The same remark applies to William Leybourn who, after speaking of Seth Partridge’s slide rule, returns to forms of Gunter’s scale, saying:[14]

There is yet another way of disposing of this Line of Proportion, by having one Line of the full length of the Ruler, and another Line of the same Radius broken in two parts between 3 and 4; so that in working your Compasses never go off of the Line: This is one of the best contrivances, but here Compasses must be used. These are all the Contrivances that I have hitherto seen of these Lines: That which I here speak of, and will shew how to use, is only two Lines of one and the same Radius, being set upon a plain Ruler of any length (the larger the better) having the beginning of one Line, at the end of the other, the divisions of each Line being set so close together, that if you find any number upon one of the Lines, you may easily see what number stands against it on the other Line. This is all the Variation. . . .

Example 1. If a Board be 1 Foot 64 parts broad, how much in length of that Board will make a Foot Square? Look upon one of your Lines (it matters not which) for 1 Foot 64 parts, and right against it on the other Line you shall find 61; and so many parts of a Foot will make a Foot square of that Board.

This contrivance solves the equation 1.64x=1, yielding centesimal parts of a foot.

James Atkinson[15] speaks of “Gunter’s scale” as “usually of Boxwood . . . commonly 2 ft. long, 1½ inch broad” and “of two kinds: long Gunter or single Gunter, and the sliding Gunter. It appears that during the seventeenth century (and long after) the Gunter’s scale was a rival of the slide rule.

III. RICHARD DELAMAIN’S GRAMMELOGIA

We begin with a brief statement of the relations between Oughtred and Delamain. At one time Delamain, a teacher of mathematics in London, was assisted by Oughtred in his mathematical studies. In 1630 Delamain published the Grammelogia, a pamphlet describing a circular slide rule and its use. In 1631 he published another tract, on the Horizontall Quadrant.[16] In 1632 appeared Oughtred’s Circles of Proportion[17] translated into English from Oughtred’s Latin manuscript by another pupil, William Forster, in the preface of which Forster makes the charge (without naming Delamain) that “another . . . went about to pre-ocupate” the new invention. This led to verbal disputes and to the publication by Delamain of several additions to the Grammelogia, describing further designs of circular slide rules and also stating his side of the bitter controversy, but without giving the name of his antagonist. Oughtred’s Epistle was published as a reply. Each combatant accuses the other of stealing the invention of the circular slide rule and the horizontal quadrant.

The two title-pages of the edition of the Grammelogia in the British Museum in London which we have called “Grammelogia IV.”

Different editions or impressions

There are at least five different editions, or impressions, of the Grammelogia which we designate, for convenience, as follows:

Grammelogia I, 1630. One copy in the Cambridge University Library.[18]

Grammelogia II, I have not seen a copy of this.

Grammelogia III, One copy in the Cambridge University Library.[19]

Grammelogia IV, One copy in the British Museum, another in the Bodleian Library, Oxford.[20]

Grammelogia V, One copy in the British Museum.

In Grammelogia I the first three leaves and the last leaf are without pagination. The first leaf contains the title-page; the second leaf, the dedication to the King and the preface “To the Reader;” the third leaf, the description of the Mathematical Ring. Then follow 22 numbered pages. Counting the unnumbered pages, there are altogether 30 pages in the pamphlet. Only the first three leaves of this pamphlet are omitted in Grammelogia IV and V.

In Grammelogia III the Appendix begins with a page numbered 52 and bears the heading “Conclusion;” it ends with page 68, which contains the same two poems on the mathematical ring that are given on the