It consisteth of two rulers of bra.s.se about 32 ynches of length, which also are halfe an ynch broad, and a quarter of an ynch thick ... At one end of both those rulers are two little sockets of bra.s.se fastened on strongly: by which the rulers are held together, and made to move one upon another, and to bee drawne out unto any length, as occasion shall require: and when you have them at the just length, there is upon one of the sockets a long Scrue-pin to scrue them fast.

There are graduations on three sides of the rulers, one graduation being the logarithmic line of numbers. He says (p. 39), "the maner of computing the Gauge-divisions I have concealed." W. Robinson, who was a friend of Oughtred, wrote him as follows:[28]

I have light upon your little book of artificial gauging, wherewith I am much taken, but I want the rod, neither could I get a sight of one of them at the time, because Mr. Allen had none left ... I forgot to ask Mr. Allen the price of one of them, which if not much I would have one of them." Oughtred annotated this pa.s.sage thus: "Or in wood, if any be made in wood by Thompson or any other."

Another of Oughtred's admirers, Sir Charles Cavendish, wrote, on February 11, 1635 thus:[29]

I thank you for your little book, but especially for the way of calculating the divisions of your gauging rod. I wish, both for their own sakes and yours, that the citizens were as capable of the acuteness of this invention, as they are commonly greedy of gain, and then I doubt not but they would give you a better recompense than I doubt now they will.

On April 20, 1638, we find Oughtred giving Elias Allen directions[30]

"about the making of the two rulers." As in 1633,[31] so now, Oughtred takes one ruler longer than the other. This 1633 instrument was used also as "a crosse-staffe to take the height of the Sunne, or any Starre above the Horizon, and also their distances." The longer ruler was called staffe, the shorter transversarie. While in 1633 he took the lengths of the two in the ratio "almost 3 to 2," in 1638, he took "the transversary three quarters of the staff's length, ... that the divisions may be larger."

VII. OTHER SEVENTEENTH CENTURY SLIDE RULES

In my History of the Slide Rule I treat of Seth Partridge, Thomas Everard, Henry Coggeshall, W. Hunt and Sir Isaac Newton.[32] Of Partridge's Double Scale of Proportion, London, I have examined a copy dated 1661, which is the earliest date for this book that I have seen. As far as we know, 1661 is the earliest date of publications on the slide rule, since Oughtred and Delamain. But it would not be surprising if the intervening 28 years were found not so barren as they seem at present. The 1661 and 1662 impressions of Partridge are identical, except for the date on the t.i.tle-page. William Leybourn, who printed Partridge's book, speaks in high appreciation of it in his own book.[33]

In 1661 was published also John Brown's first book, Description and Use of a Joynt-Rule, previously mentioned. In Chapter XVIII he describes the use of "Mr. Whites rule" for the measuring of board and timber, round and square. He calls this a "sliding rule." The existence, in 1661, of a "Whites rule" indicates activities in designing of which we know as yet very little. In his book of 1761, previously quoted, Brown gives a drawing of "White's sliding rule" (p. 193); also a special contrivance of his own, as indicated by him in these words:

A further improvement of the Triangular Quadrant, as I have made it several times, with a sliding Cover on the in-side, when made hollow, to carry Ink, Pens, and Compa.s.ses; then on the sliding Cover, and Edges, is put the Line of Numbers, according to Mr. White's first Contrivance for manner of operation; but much augmented, and made easie, by John Brown.

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 Compa.s.ses.

One line on Coggeshall's rule begins with 4 and extends to 40, these numbers being the "Girt" (a quarter of the circ.u.mference), 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 t.i.tle 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-geometric.u.m, 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."

The data presented in this article show that, while the earliest slide rules were of the circular type, the later slide rules of the seventeenth century were of the rectilinear type.[39]

January 12, 1915.

Footnotes

[1]F. Cajori, History of the Logarithmic Slide Rule and Allied Instruments, New York, 1909, pp. 7-14, also Addenda i-vi.

[2]F. Cajori, "On the Invention of the Slide Rule," in Colorado College Publication, Engineering Series Vol. 1, 1910. An abstract of this is given in Nature (London), Vol. 82, 1909, p. 267.

[3]F. Cajori, History etc., p. 14.

[4]Art. "Slide Rule" in the Penny Cyclopaedia and in the English Cyclopaedia [Arts and Sciences].

[5]Anthony Wood, Athenae oxonienses (Ed. P. Bliss), London, Vol. III, 1817, p. 423.

[6]The full t.i.tle of the book which Wingate published on this subject in Paris is as follows:

L'Vsage

de la

Reigle de

Proportion

en l'Arithmetique &

Geometrie.

Par Edmond Vvingate,

Gentil-homme Anglois.

??? ?? f??ea???, ?s? ?s? p???a???.

In tenui, sed n tenuis vsusve, laborne.

A Paris,

Chez Melchior Mondiere,

demeurant en l'Isle du Palais,

a la

rue de Harlay aux deux Viperes.

M. DC. XXIV.

Auec Priuilege du Roy.

Back of the t.i.tle page is the announcement:

Notez que la Reigle de Proportion en toutes facons se vend a Paris chez Melchior Tauernier, Graueur & Imprimeur du Roy pour les Tailles douces, demeurant en l'Isle du Palais sur le Quay qui regarde la Megisserie a l'Espic d'or.

[7]The t.i.tle-page of the edition of 1658 is as follows:

The Use of the Rule of Proportion in Arithmetick & Geometrie. First published at Paris in the French tongue, and dedicated to Monsieur, the then king's onely Brother (now Duke of Orleance). By Edm. Wingate, an English Gent. And now translated into English by the Author. Whereinto is now also inserted the Construction of the same Rule, & a farther use thereof ... 2nd edition inlarged and amended. London, 1658.

[8]Memories of the Life of that Learned Antiquary, Elias Ashmole, Esq.; Drawn up by himself by way of Diary. With Appendix of original Letters.

Publish'd by Charles Burman, Esq., London, 1717, p. 23.

[9]Mathematical Tables, 1811, p. 36, and art. "Gunter's Line" in his Phil.

and Math. Dictionary, London, 1815.

[10]To the English Gentrie, and all others studious of the Mathematicks, which shall bee readers hereof. The just Apologie of Wil: Ovghtred, against the slaunderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum projectio circularis. We shall refer to this doc.u.ment as Epistle. It was published without date in 32 unnumbered pages of fine print, and was bound in with Oughtred's Circles of Proportion, in the editions of 1633 and 1639. In the 1633 edition it is inserted at the end of the volume just after the Addition vnto the Vse of the Instrument etc., and in that of 1639 immediately after the preface. It was omitted from the Oxford edition of 1660. The Epistle was also published separately. There is a separate copy in the British Museum, London. Aubrey, in his Brief Lives, edited by A. Clark, Vol. II, Oxford, 1898, p. 113, says quaintly, "He writt a st.i.tch't pamphlet about 163(?4) against ... Delamaine."

[11]Thomas Browne is mentioned by Stone in his Mathematical Instruments, London 1723, p. 16. See also Cajori, History of the Slide Rule, New York, 1909, p. 15.

[12]The Description and Use of a Joynt-Rule: ... also the use of Mr.

White's Rule for measuring of Board and Timber, round and square; With the manner of Vsing the Serpentine-line of Numbers, Sines, Tangents, and Versed Sines. By J. Brown, Philom., London, 1661.