VII — Other seventeenth century slide rules — On the History of Gunter’s Scale and the Slide Rule by Florian Cajori

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.

Footnotes

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

2F. 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.

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

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

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

6The full title of the book which Wingate published on this subject in Paris is as follows:

Εἂν ἧς φιλεμαθὴς, ἕση ἥση πολυμαθὴς.

In tenui, sed nõ tenuis vsusve, laborne. |

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.

7The 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.

8Memories 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.

9Mathematical Tables, 1811, p. 36, and art. “Gunter’s Line” in his Phil. and Math. Dictionary, London, 1815.

10To 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.”

11Thomas 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.

12The 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.

13A 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.

14William Leybourn, op. cit., pp. 129, 130, 132, 133.

15James 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.

16R. Delamain, The Making, Description, and Use of a small portable Instrument... called a Horizontall Quadrant, etc., London, 1631.

17Oughtred’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.

18The full title of the Grammelogia I is as follows:

19Grammelogia III is the same as Grammelogia I, except for the addition of an appendix, entitled:

... 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.

20Grammelogia 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. |

There is no date. There follows the diagram of a second circular slide rule, with the inscription within the innermost ring: Typus proiectionis Annuli adaucti vt in Conslusione Lybri praelo commissi, Anno 1630 promisi. There are numerous drawings in the Grammelogia, all of which, excepting the drawings of slide rules on the engraved title-pages of Grammelogia IV and V, were printed upon separate pieces of paper and then inserted by hand into the vacant spaces on the printed pages reserved for them. Some drawings are missing, so that the Bodleian Grammelogia IV differs in this respect slightly from the copy in the British Museum and from the British Museum copy of Grammelogia V.

21Epistle, p. (8).

22Aubrey, op. cit., Vol. II., p. 111.

23Rigaud, Correspondence of Scientific Men during the 17th Century, Vol. I, Oxford, 1841, p. 11.

24Dictionary of National Biography, Art. “Delamain, Richard.” See also Rev. Charles J. Robinson, Taylors’ School, from A.D. 1562 to 1874, Vol. I, 1882, p. 151; Journal of the House of Commons, Vol. IV., p. 197b; Sixth Report of the Royal Commission on Historical Manuscripts, Part I, Report and Appendix, London, 1877. In this Appendix, p. 82, we read the following:

Oct. 22 [1645] Petition of Sarah Delamain, relict of Richard Delamain. Petitioner’s husband was servant to the King, and one of His Majesty’s engineers for the fortification of the kingdom, and his tutor in mathematical arts; but upon the breaking out of the war he deserted the Court, and was called by the State to several employments, in fortifying the towns of Northampton, Newport, and Abingdon; and was also abroad with the armies as Quartermaster-General of the Foot, and therein died. Petitioner is left a disconsolate widow with ten children, the four least of whom are now afflicted with sickness, and petitioner has nothing left to support them. There are several considerable sums of money due to the petitioner, as well from the King as the State. Prays that she may have some relief amongst other widows. See L. J., VII. 6. 657.

25Anthony Wood, Athenae Oxonienses (Edition Bliss) Vol. IV., London, 1820, p. 34.

26The New Artificial Gauging Line or Rod: together with rules concerning the use thereof: Invented and written by WILLIAM OUGHTRED, etc., London, 1633. The copy we have seen is in the Bodleian Library, Oxford. The book is small sized and has 40 pages.

[27]Oughtred, op. cit., p. 11.

28S. J. Rigaud, Correspondence of Scientific Men of the 17th Century, Oxford, Vol. I, 1841, p. 17.

29Rigaud, loc. cit., p. 22.

30Rigaud, loc. cit., pp. 30, 31.

31Oughtred, An Addition vnto the Vse of the Instrument called the Circles of Proportion, London, 1633, p. 63.

32F. Cajori, History of the Slide Rule, New York, 1909, pp. 16-22, Addenda, pp. vi-ix.

33W. Leybourn, op. cit., 1673, Preface, and pp. 128-29.

34Cajori op. cit., Addenda, p. ix.

35William Leybourn, op. cit., 1673, p. 35.

36See Cajori, op. cit., pp. 20, 28, Addenda, p. ix.

37See F. Cajori, “A Note on the History of the Slide Rule,” Bibliotheca mathematica, 3 F., Vol. 10, pp. 161-163.

38John Atkinson, op. cit., 1694, p. 204.

39Probably the oldest slide rule now in existence is owned by St. John’s College, Oxford, and is in the form of a brass disc, 1 ft. 6 in. in diameter. It was exhibited along with other instruments in May, 1919. According to the Catalogue of a Loan Exhibition of Early Scientific Instruments in Oxford, opened May 16, 1919, the instrument is inscribed with the name of the maker (“Elias Allen fecit”) and with the name of the donor, Georgius Barkham. It is dated 1635, which is only three years after the first publication of Oughtred’s description of his circular slide rule. It is stated in the Catalogue: “Unfortunately all the movable parts but the base-plate and a couple of thumb-screws are missing. The face of the instrument is engraved with Oughtred’s Horizontal Instrument. The back is engraved with eleven Circles of Proportion as described in Arthur Haughton’s book, a copy of which was presented to St. John’s College by George Barkham, to explain the use of the instrument.” As Arthur Haughton’s Oxford edition of Oughtred’s Circles of Proportion did not appear until 1660, it would seem that the instrument was probably not presented to the College before 1660. As far as is known, the next oldest slide rule is of the year 1654, kept in the South Kensington Museum, London, and is described in Nature of March 5, 1914. It is a rectilinear rule, “of boxwood, well made, and bound together with brass at the two ends. It is of the square type, a little more than 2 ft. in length, and bears the logarithmic lines first described by Edmund Gunter. Of these, the num, sin and tan lines are arranged in pairs, identical and contiguous, one line in each pair being on the fixed part, and the other on the slide.” The instrument is inscribed, “Made by Robert Bissaker for T. W., 1654.” Nowhere else have we seen reference to Robert Bissaker. His slide rule seems to antedate the “Whites rule” mentioned above. [This foot-note was added on October 15, 1919.]

Notes

2. Constructive Theory of the Unicursal Plane Quartic by Synthetic Methods, by Annie Dale Biddle. Pp. 27-54, 31 text-figures. September, 1912.50

3. A Discussion by Synthetic Methods of Two Projective Pencils of Conics, by Baldwin Munger Woods. Pp. 55-85. February, 1913.50

4. A Complete Set of Postulates for the Logic of Classes Expressed in Terms of the Operation “Exception”, and a Proof of the Independence of a Set of Postulates due to Del Ré, by B. A. Bernstein. Pp. 87-96. May, 1914.10

5. On a Tabulation of Reduced Binary Quadratic Forms of a Negative Determinant, by Harry N. Wright. Pp. 97-114. June, 1914.20

6. The Abelian Equations of the Tenth Degree Irreducible in a Given Domain of Rationality, by Charles G. P. Kuschke. Pp. 115-162. June, 1914.50

7. Abridged Tables of Hyperbolic Functions, by F. E. Pernot. Pp. 163-169. February, 1915.10

8. A List of Oughtred’s Mathematical Symbols, with Historical Notes, by Florian Cajori. Pp. 171-186. February, 1920.25

9. On the History of Gunter’s Scale and the Slide Rule during the Seventeenth Century, by Florian Cajori. Pp. 187-209. February, 1920.35

10. On a Birational Transformation Connected with a Pencil of Cubics, by Arthur Robinson Williams. Pp. 211-222. February, 1920.15

11. Classification of Involutory Cubic Space Transformations, by Frank Ray Morris. Pp. 223-240. February, 1920.25

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