On the History of Gunter's Scale and the Slide Rule during the Seventeenth Century

Oughtred said that he had no desire to publish his invention, but in the vacation of 1630 finally promised William Forster to let him bring out a translation. Oughtred claims that Delamain got the invention from him at Alhallontide [November 1], 1630, when they met in London. The accounts of that meeting we proceed to give in double column.

Delamain’s Statement Grammelogia IV, page (98)

“.. about Alhalontide 1630. (as our Authors reporteth) was the time he was circumvented, and then his intent in a loving manner (as before) he opened unto me, which particularly I will dismantle in the very naked truth: for, wee being walking together some few weekes before Christmas, upon Fishstreet hill, we discoursed upon sundry things Mathematicall, both Theoreticall and Practicall, and of the excellent inventions and helpes that in these dayes were produced, amongst which I was not a little taken with that of the Logarythmes, commending greatly the ingenuitie of Mr. Gunter in the Projection, and inventing of his Ruler, in the lines of proportion, extracted from these Logarythmes for ordinary Practicall uses; He replyed unto me (in these very words) What will yov say to an Invention that I have, which in a lesse extent of the Compasses shall worke truer then that of Mr. Gunters Ruler, I asked him then of what forme it was, he answered with some pause (which no doubt argued his suspition of mee that I might conceive it) that it was Arching-wise, but now hee sayes that hee told mee then, it was Circular (but were I put to my oath to avoid the guilt of Conscience I would conclude in the former.) At which immediately I answered, I had the like my selfe, and so we discoursed not a word more touching that subject.. Then after my coming home I sent him a sight of my Projection drawne in Pastboard: Now admit I had not the Invention of my Ring before I discoursed.. it was not so facil for mee.. to raise and compose so complete, and absolute an Instrument from so small a principle, or glimpse of light.”

Oughtred’s Statement Epistle, page (23)

“Shortly after my gift to Elias Allen, I chanced to meet with Richard Delamain in the street (it was at Alhallontide) and as we walked together I told him what an Instrument I had given to Master Allen, both of the Logarithmes projected into circles, which being lesse then one foot diameter would performe as much as one of Master Gunters Rulers of sixe feet long: and also of the Prostaphaereses of the Plannets and second motions. Such an invention have I said he: for now his intentions (that is his ambition) beganne to worke:.. But he saith, Then after my comming home I sent him a sight of my projection drawne in past-board. See how notoriously he jugleth without an Instrument. Then after: how long after? a sight of my projection: of how much? More then seven weekes after on December 23, he sent to mee the line of numbers onely set upon a circle:.. and so much onely he presented to his Majesty: but as for Sine or tangent of his, there was not the least shew of any. Neither could he give to Master Allen any direction for the composure of the circles of his Ring, or for the division of them: as upon his oath Master Allen will testify how hee misled him, and made him labour in vain above three weeks together, until Master Allen himselfe found out his ignorance and mistaking, which is more cleare then is possible with any impudence to be outfaced.”

Oughtred makes a further statement (Epistle, p. (24)) as follows:

Delamain hearing that Brown with his Serpentine had another line by which he could worke to minutes in the 90 degree of sines.. gave the [his] booke to Browne: who in thankfulnesse could not but gratify Delamain with his Lines also: and teach him the use of them, but especially of the great Line: with this caution on both sides, that one should not meddle with the others invention. Two dayes after Delamain.. because he had found some things to be altered therin… asked for the booke.. but as soone as he had got it in his hands he rent out all the middle part with the two Schemes & put them up in his pocket & went his way.. and.. laboureth to recall all the bookes he had given forth.. And shortly after this he got a new Printer (who was ignorant of his former Schemes) to print him new: giving him an especiall charge of the outermost line newly graven in the Plate, which indeed is Brownes very line: and then altering his book.

This and other statements made by Oughtred seem damaging to Delamain’s reputation. But it is quite possible that Oughtred’s guesses as to Delamain’s motives are wrong. Moreover, some of Oughtred’s statements are not first hand knowledge with him, but mere hearsay. One may accept his first hand facts and still clear Delamain of wrong doing. There is always danger that rival claimants of an invention or discovery will proceed on the assumption that no one else could possibly have come independently upon the same devices that they themselves did; the history of science proves the opposite. Seldom is an invention of any note made by only one man. We do not feel competent to judge Delamain’s case. We know too little about him as a man. We incline to the opinion that the hypothesis of independent invention is the most plausible. At any rate, Delamain figures in the history of the slide rule as the publisher of the earliest book thereon and as an enthusiastic and skillful designer of slide rules.

The effect of this controversy upon interested friends was probably small. Doubtless few people read both sides. Oughtred says:21 “this scandall.. hath with them, to whom I am not knowne, wrought me much prejudice and disadvantage.” Aubrey,22 a friend of Oughtred, refers to Delamain “who was so sawcy to write against him” and remembers having seen “many yeares since, twenty or more good verses made” against Delamain. Another friend of Oughtred, William Robinson, who had seen some of Delamain’s publications, but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before the appearance of the latter’s Epistle:

I cannot but wonder at the indiscretion of Rich. Delamain, who being conscious to himself that he is but the pickpurse of another man’s wit, would thus inconsiderately provoke and awake a sleeping lion.. he hath so weakly (though in my judgment, vaingloriously enough) commended his own labour.23

Delamain presented King Charles I with one of his sun-dials, also with a manuscript and, later, with a printed copy of his book of 1630. A drawing of his improved slide rule was sent to the King and the Grammelogia IV is dedicated to him. The King must have been favorably impressed, for Delamain was appointed tutor to the King in mathematics. His widow petitioned the House of Lords in 1645 for relief; he had ten children.24

Anthony Wood states that Charles I, on the day of his execution, commanded his friend Thomas Herbert “to give his son the duke of York his large ring-sundial of silver, a jewel his maj. much valued.” Anthony Wood adds, “it was invented and made by Rich. Delamaine a very able mathematician, who projected it, and in a little printed book did shew its excellent use in resolving many questions in arithmetic and other rare operations to be wrought by it in the mathematics.”25

VI. OUGHTRED’S GAUGING LINE, 1633

It has not been generally known, hitherto, that Oughtred designed a rectilinear slide rule for gauging and published a description thereof in 1633.26 In his Circles of Proportion, chapter IX, Oughtred had offered a closer approximation than that of Gunter for the capacity of casks. The Gauger of London expostulated with Oughtred for presuming to question anything that Gunter had written. The ensuing discussion led to an invitation extended by the Company of Vintners to the instrument maker Elias Allen to request Oughtred to design a gauging rod.27 This he did, and Allen received an order for “threescore” instruments. On page 19 Oughtred describes his ‘Gauging Rod:’

It consisteth of two rulers of brasse 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 brasse 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 passage 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 directions30 “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 title-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 Compasses; 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 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 description37 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.”

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.

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 title 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. |

Εἂν ἧς φιλεμαθὴς, ἕση ἥση πολυμαθὴς.

In tenui, sed nõ tenuis vsusve, laborne. |

A Paris, | Chez Melchior Mondiere, | demeurant en l’Isle du Palais, | à la | ruë de Harlay aux deux Viperes. | M. DC. XXIV. | Auec Priuilege du Roy. |

Back of the title page is the announcement:

Notez que la Reigle de Proportion en toutes façons se vend à 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 à l’Espic d’or.

7

The title-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 document 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 stitch’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.

13

A Collection of Centers and Useful Proportions on the Line of Numbers, by John Brown, 1662(?), 16 pages; Description and Use of the Triangular Quadrant, by John Brown, London, 1671; Wingate’s Rule of Proportion in Arithmetick and Geometry: or Gunter’s Line. Newly rectified by Mr. Brown and Mr. Atkinson, Teachers of the Mathematicks, London, 1683; The Description and Use of the Carpenter’s-Rule: Together with the Use of the Line of Numbers commonly call’d Gunter’s-Line, by John Brown, London, 1704.

14

William Leybourn, op. cit., pp. 129, 130, 132, 133.

15

James Atkinson’s edition of Andrew Wakely’s The Mariners Compass Rectified, London, 1694 [Wakely’s preface dated 1664, Atkinson’s preface, 1693]. Atkinson adds An Appendix containing Use of Instruments most useful in Navigation. Our quotation is from this Appendix, p. 199.

16

R. Delamain, The Making, Description, and Use of a small portable Instrument.. called a Horizontall Quadrant, etc., London, 1631.

17

Oughtred’s description of his circular slide rule of 1632 and his rectilinear slide rule of 1633, as well as a drawing of the circular slide rule, are reproduced in Cajori’s History of the Slide Rule, Addenda, pp. ii-vi.

18

The full title of the Grammelogia I is as follows:

Gram̄elogia | or, | The Mathematicall Ring. | Shewing (any reasonable Capacity that hath | not Arithmeticke) how to resolve and worke | all ordinary operations of Arithmeticke. | And those which are most difficult with greatest | facilitie: The extraction of Roots, the valuation of | Leases, &c. The measuring of Plaines | and Solids. | With the resolution of Plaine and Sphericall | Triangles. | And that onely by an Ocular Inspection, | and a Circular Motion. | Naturae secreta tempus aperit. | London printed by John Haviland, 1630.

19

Grammelogia III is the same as Grammelogia I, except for the addition of an appendix, entitled:

De la Mains | Appendix | Vpon his | Mathematicall | Ring. Attribuit nullo (praescripto tempore) vitae | vsuram nobis ingeniique Deus. | London, |

.. The next line or two of this title-page which probably contained the date of publication, were cut off by the binder in trimming the edges of this and several other pamphlets for binding into one volume.

20

Grammelogia IV has two title pages. The first is Mirifica Logarithmoru’ Projectio Circularis. There follows a diagram of a circular slide rule, with the inscription within the innermost ring: Nil Finis, Motvs, Circvlvs vllvs Habet. The second title page is as follows:

Grammelogia | Or, the Mathematicall Ring. | Extracted from the Logarythmes, and projected Circular: Now published in the | inlargement thereof unto any magnitude fit for use: shewing any reason- | able capacity that hath not Arithmeticke how to resolve and worke, | all ordinary operations of Arithmeticke: | And those that are most difficult with greatest facilitie, the extracti- | on of Rootes, the valuation of Leases, &c. the measuring of Plaines and Solids, | with the resolution of Plaine and Sphericall Triangles applied to the | Practicall parts of Geometrie, Horologographie, Geographie | Fortification, Navigation, Astronomie, &c. | And that onely by an ocular inspection, and a Circular motion, Invented and first published, by R. Delamain, Teacher, and Student of the Mathematicks. | Naturae secreta tempus aperit. |