He gives no drawing of his “triangular quadrant,” hence his account of it is unsatisfactory. He explains the use of “gage-points.” His placing logarithmic lines on the edges of instrument boxes was outdone in oddity later by Everard who placed them on tobacco-boxes.[34] In Brown’s publication of 1704 the White slide rule is given again, “being as neat and ready a way as ever was used.” He tells also of a “glasier’s sliding rule.” William Leybourn explains in 1673 how Wingate’s double and triple lines for squaring and cubing, or square and cube root, can be used on slide rules.[35]
Beginning early in the history of the slide rule, when Oughtred designed his “gauging rod,” we notice the designing of rules intended for very special purposes. Another such contrivance, which enjoyed long popularity, was the Timber Measure by a Line, by Hen. Coggeshall, Gent., London, 1677, a booklet of 35 pages. Coggeshall says in his preface:
For what can be more ready and easie, then having set twelve to the length, to see the Content exactly against the Girt or Side of the Square. Whereas on Mr. Partridge’s Scale the Content is the Sixth Number, which is far more troublesome then [even] with Compasses.
One line on Coggeshall’s rule begins with 4 and extends to 40, these numbers being the “Girt” (a quarter of the circumference), which in ordinary practice of measuring round timber lies between 4 inches and 40 inches. This “Girt line” slides “against the line of Numbers in two Lengths, to which it is exactly equal.” A second edition, 1682, shows some changes in the rule, as well as an enlargement and change of title of the book itself: A Treatise of Measures, by a Two-foot Rule, by H. C. Gent, London, 1682. In this, the description of the rule is given thus:
There are four Lines on each flat of this Rule; two next the outward edges, which are Lines of Measure; and two next the inward edges, which are Lines of Proportion. On one flat, next the inward edges, is the Square-line [Girt-line in round timber measurement] with the Line of Numbers his fellow. Next the outward, a Line of Inches divided into Halfs, Quarters, and Half-Quarters; from 1 to 12 on one Rule; and from 12 to 24 on the other. On the other flat, next the inward edges, is the double Scale of Numbers [for solving proportions]. Next the outward on one Rule a Line of Inches divided each into ten parts; and this for gauging, etc. On the other a foot divided into 100 parts.
Later further changes were introduced in Coggeshall’s rule.[36]
It is worthy of note that Coggeshall’s slide rule book, The Art of Practical Measuring, was reviewed in the Acta eruditorum, anno 1691, p. 473; hence Leupold’s description[37] of the rectilinear slide rule in his Theatrum arithmetico-geometricum, Leipzig, 1727, Cap. XIII, p. 71, is not the earliest reference to the rectilinear rule found in German publications. The above date is earlier even than Biler’s reference to a circular slide rule in his Descriptio instrumenti mathematici universalis of 1696.
Two noted slide rules for gauging were described by Tho. Everard, Philomath, in his Stereometry made easie, London, 1684. He designates his lines by the capital letters A, B, C, D, E. On the first instrument, A on the rule, and B and C on the slide, have each two radiuses of numbers, D has only one, while E has three. The second rule is described in an Appendix; it is one foot long, with two slides enabling the rule to be extended to 3 feet.
Everard’s instruments were made in London by Isaac Carver who, soon after, himself wrote a sixteen-page Description and Use of a New Sliding Rule, projected from the Tables in the Gauger’s Magazine, London, 1687, which was “printed for William Hunt” and bound in one volume with a book by Hunt, called The Gauger’s Magazine, London, 1687. This appears to be the same William Hunt who later brought out descriptions of his own of slide rules. The instrument described by Carver “consists of three pieces, two whereof are moveable to be drawn out till the whole be 36 inches long.” It has several non-logarithmic graduations, together with logarithmic lines marked A, B, C, D, of which A, B, C are “double lines,” and D a “single line” used for squares and square roots. It is designed for the determination of the vacuity of a “spheroidal cask lying,” a “spheroidal cask standing,” and a “parabolical cask lying.”
Another seventeenth century writer on the slide rule is John Atkinson, whom we have mentioned earlier. He says:[38] “The Lines of Numbers, Sines and Tangents, are set double, that is, one on each side, as the middle piece slides: which middle piece is so contrived, to slip to and fro easily, to slide out, and to be put in any side uppermost, in order to bring those Lines together (or against one another) most proper for solving the Question, wrought by Sliding-Gunter.”