Chapter I: Front Matter (1)

UNIVERSITY OF CALIFORNIA PUBLICATIONS
IN
MATHEMATICS

Vol. 1, No. 9, pp. 187-209 February 17, 1920

ON THE HISTORY OF GUNTER’S SCALE AND
THE SLIDE RULE DURING THE
SEVENTEENTH CENTURY

BY
FLORIAN CAJORI

UNIVERSITY OF CALIFORNIA PRESS
BERKELEY

TABLE OF CONTENTS

PAGE
I. Introduction 187
II. Innovations in Gunter’s Scale 188
Changes introduced by Edmund Wingate 188
Changes introduced by Milbourn 189
Changes introduced by Thomas Brown and John Brown 190
Changes introduced by William Leybourn 192
III. Richard Delamain’s “Grammelogia” 192
Different editions or impressions 194
Description of Delamain’s instrument of 1630 195
Delamain’s later designs, and directions for using his
instruments 197
IV. Controversy between Oughtred and Delamain on the invention of
the circular slide rule 199
V. Independence and priority of invention 203
VI. Oughtred’s “Gauging Line,” 1633 206
VII. Other seventeenth century slide rules 207

I. INTRODUCTION

In my history of the slide rule[1], and my article on its invention[2] it is shewn that William Oughtred and not Edmund Wingate is the inventor, that Oughtred’s circular rule was described in print in 1632, his rectilinear rule in 1633. Richard Delamain is referred to as having tried to appropriate the invention to himself[3] and as having written a scurrilous pamphlet against Oughtred. All our information about Delamain was taken from De Morgan,[4] who, however, gives no evidence of having read any of Delamain’s writings on the slide rule. Through Dr. Arthur Hutchinson of Pembroke College, Cambridge, I learned that Delamain’s writings on the slide rule were available. In this article will be given: First, some details of the changes introduced during the seventeenth century in the design of Gunter’s scale by Edmund Wingate, Milbourn, Thomas Brown, John Brown and William Leybourn; second, an account of Delamain’s book of 1630 on the slide rule which antedates Oughtred’s first publication (though Oughtred’s date of invention is earlier than the date of Delamain’s alleged invention) and of Delamain’s later designs of slide rules; third, an account of the controversy between Delamain and Oughtred; fourth, an account of a later book on the slide rule written by William Oughtred, and of other seventeenth century books on the slide rule.

II. INNOVATIONS IN GUNTER’S SCALE

Changes introduced by Wingate

We begin with Anthony Wood’s account of Wingate’s introduction of
Gunter’s scale into France.[5]

In 1624 he transported into France the rule of proportion, having a
little before been invented by Edm. Gunter of Gresham Coll. and
communicated it to most of the chiefest mathematicians then residing in
Paris: who apprehending the great benefit that might accrue thereby,
importun’d him to express the use thereof in the French tongue. Which
being performed accordingly, he was advised by monsieur Alleawne the
King’s chief engineer to dedicate his book to monsieur the King’s only
brother, since duke of Orleans. Nevertheless the said work coming forth
as an abortive (the publishing thereof being somewhat hastened, by reason
an advocate of Dijon in Burgundy began to print some uses thereof, which
Wingate had in a friendly way communicated to him) especially in regard
Gunter himself had learnedly explained its use in a far larger volume.[6]

Gunter’s scale, which Wingate calls the “rule of proportion,” contained, as described in the French edition of 1624, four lines: (1) A single line of numbers; (2) a line of tangents; (3) a line of sines; (4) a line, one foot in length, divided into 12 inches and tenths of inches, also a line, one foot in length, divided into tenths and hundredths.

The English editions of this book which appeared in 1623 and 1628 are devoid of interest. The editions of 1645 and 1658 contain an important innovation.[7] In the preface the reasons why this instrument has not been used more are stated to be: (1) the difficulty of drawing the lines with exactness, (2) the trouble of working thereupon by reason (sometimes) of too large an extent of the compasses, (3) the fact that the instrument is not readily portable. The drawing of Wingate’s arrangement of the scale in the editions of 1645 and 1658 is about 66 cm. (26.5 in.) long. It contains five parallel lines, about 66 cm. long, each having the divisions of one line marked on one side and of another line on the other side. Thus each line carries two graduations: (1) A single logarithmic line of numbers; (2) a logarithmic line of numbers thrice repeated; (3) the first scale repeated, but beginning with the graduations which are near the middle of the first scale, so that its graduation reads 4, 5, 6, 7, 8, 9, 1, 2, 3; (4) a logarithmic line of numbers twice repeated; (5) a logarithmic line of tangents; (6) a logarithmic line of sines; (7) the rule divided into 1000 equal parts; (8) the scale of latitudes; (9) a line of inches and tenths of inches; (10) a scale consisting of three kinds, viz., a gauge line, a line of chords, and a foot measure, divided into 1000 equal parts.

Important are the first and second scales, by which cube root extraction was possible “by inspection only, without the aid of pen or compass;” similarly the third and fourth scales, for square roots. This innovation is due to Wingate. The 1645 edition announces that the instrument was made in brass by Elias Allen, and in wood by John Thompson and Anthony Thompson in Hosier Lane.

Changes introduced by Milbourn

William Leybourn, in his The Line of Proportion or Numbers, Commonly called Gunter’s Line, Made Easie, London, 1673, says in his preface “To the Reader:”

The Line of Proportion or Numbers, commonly called (by Artificers)
Gunter’s Line, hath been discoursed of by several persons, and variously
applied to divers uses; for when Mr. Gunter had brought it from the
Tables to a Line, and written some Uses thereof, Mr. Wingate added divers
Lines of several lengths, thereby to extract the Square or Cube Roots,
without doubling or trebling the distance of the Compasses: After him Mr.
Milbourn, a Yorkshire Gentleman, disposed it in a Serpentine or Spiral
Line, thereby enlarging the divisions of the Line.

On pages 127 and 128 Leybourn adds:

Again, One T. Browne, a Maker of Mathematical Instruments, made it in a
Serpentine or Spiral Line, composed of divers Concentrick Circles,
thereby to enlarg the divisions, which was the contrivance of one Mr.
Milburn a Yorkshire Gentleman, who writ thereof, and communicated his
Uses to the aforesaid Brown, who (since his death) attributed it to
himself: But whoever was the contriver of it, it is not without
inconvenience; for it can in no wise be made portable; and besides
(instead of compasses) an opening Joynt with thirds [threads] must be
placed to move upon the Centre of the Instrument, without which no
proportion can be wrought.

This Mr. Milburn is probably the person named in the diary of the antiquarian, Elias Ashmole, on August 13 [1646?]; “I bought of Mr. Milbourn all his Books and Mathematical Instruments.”[8] Charles Hutton[9] says that Milburne of Yorkshire designed the spiral form about 1650. This date is doubtless wrong, for Thomas Browne who, according to Leybourn, got the spiral form of line from Milbourn, is repeatedly mentioned by William Oughtred in his Epistle[10] printed some time in 1632 or 1633. Oughtred does not mention Milbourn, and says (page 4) that the spiral form “was first hit upon by one Thomas Browne a Joyner, . . . the serpentine revolution being but two true semicircles described on severall centers.”[11]

Changes introduced by Thomas Brown and John Brown

Thomas Brown did not publish any description of his instrument, but his son, John Brown, published in 1661 a small book,[12] in which he says (preface) that he had done “as Mr. Oughtred with Gunter’s Rule, to a sliding and circular form; and as my father Thomas Brown into a Serpentine form; or as Mr. Windgate in his Rule of Proportion.” He says also that “this brief touch of the Serpentine-line I made bold to assert, to see if I could draw out a performance of that promise, that hath been so long unperformed by the promisers thereof.” Accordingly in Chapter XX he gives a description of the serpentine line, “contrived in five (or rather 15) turn.” Whether this description, printed in 1661, exactly fits the instrument as it was developed in 1632, we have no means of knowing. John Brown says:

1. First next the center is two circles divided one into 60, the other
into 100 parts, for the reducing of minutes to 100 parts, and the
contrary.

2. You have in seven turnes two inpricks, and five in divisions, the
first Radius of the sines (or Tangents being neer the matter, alike to
the first three degrees,) ending at 5 degrees and 44 minutes.

3. Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in
three margents in divisions, and the line of versed sines in pricks,
under the line of Tangents, according to Mr. Gunter’s cross-staff: the
sines and Tangents beginning at 5 degrees, and 44 minutes where the other
ended, and proceeding to 90 in the sines, and 45 in the Tangents. And the
line of numbers beginning at 10, and proceeding to 100, being one entire
Radius, and graduated into as many divisions as the largeness of the
instrument will admit, being 10 to 10 50 into 50 parts, and from 50 to
100 into 20 parts in one unit of increase, but the Tangents are divided
into single minutes from the beginning to the end, both in the first,
second and third Radiusses, and the sines into minutes; also from 30
minutes to 40 degrees, and from 40 to 60, into every two minutes, and
from 60 to 80 in every 5th minute, and from 80 to 85 every 10th, and the
rest as many as can be well discovered.

The versed sines are set after the manner of Mr. Gunter’s Cross-staff,
and divided into every 10th minutes beginning at 0, and proceeding to 156
going backwards under the line of Tangents.

4. Fourthly, beyond the Tangent of 45 in one single line, for one Turn is
the secants to 51 degrees, being nothing else but the sines reitterated
beyond 90.

5. Fifthly, you have the line of Tangents beyond 45, in 5 turnes to 85
degrees, whereby all trouble of backward working is avoided.

6. Sixthly, you have in one circle the 180 degrees of a Semicircle, and
also a line of natural sines, for finding of differences in sines, for
finding hour and Azimuth.

7. Seventhly, next the verge or outermost edge is a line of equal parts
to get the Logarithm of any number, or the Logarithm sine and Tangent of
any ark or angle to four figures besides the carracteristick.

8. Eightly and lastly, in the space place between the ending of the
middle five turnes, and one half of the circle are three prickt lines
fitted for reduction. The uppermost being for shillings, pence and
farthings. The next for pounds, and ounces, and quarters of small
Averdupoies weight. The last for pounds, shillings and pence, and to be
used thus: If you would reduce 16s. 3d. 2q. to a decimal fraction, lay
the hair or edge of one of the legs of the index on 16. 3½ in the line of
1. s. d. 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 last page of Grammelogia I but differs slightly in the spelling of some of the words. The 51 pages which must originally have preceded page 52, we have not seen. The edition containing these we have designated Grammelogia II. The reason for the omission of these 51 pages can only be conjectured. In Oughtred’s Epistle (p. 24), it is stated that Delamain had given a copy of the Grammelogia to Thomas Brown, and that two days later Delamain asked for the return of the copy, “because he had found some things to be altered therein” and “rent out all the middle part.” Delamain labored “to recall all the bookes he had given forth, (which were many) before the sight of Brownes Lines.” These spiral lines Oughtred claimed that Delamain had stolen from Brown. The title-page and page 52 are the only parts of the Appendix, as given in Grammelogia III, that are missing in the Grammelogia IV and V.

Grammelogia IV answers fully to the description of Delamain’s pamphlet contained in Oughtred’s Epistle. It was brought out in 1632 or 1633, for what appears to be the latest part of it contains a reference (page 99) to the Grammelogia I (1630) as “being now more then two yeares past.” Moreover, it refers to Oughtred’s Circles of Proportion, 1632, and Oughtred’s reply in the Epistle was bound in the Circles of Proportion having the Addition of 1633. For convenience of reference we number the two title-pages of Grammelogia IV, “page (1)” and “page (2),” as is done by Oughtred in his Epistle. Grammelogia IV contains, then, 113 pages. The page numbers which we assign will be placed in parentheses, to distinguish them from the page numbers which are printed in Grammelogia IV. The pages (44)-(65) are the same as the pages 1-22, and the pages (68)-(83) are the same as the pages 53-68. Thus only thirty-eight pages have page numbers printed on them. The pages (67) and (83) are identical in wording, except for some printer’s errors; they contain verses in praise of the Ring, and have near the bottom the word “Finis.” Also, pages (22) and (23) are together identical in wording with page (113), which is set up in finer type, containing an advertisement of a part of Grammelogia IV explaining the mode of graduating the circular rules. There are altogether six parts of Grammelogia IV which begin or end by an address to the reader, thus: “To the Reader,” “Courteous Reader,” or “To the courteous and benevolent Reader . . .,” namely the pages (8), (22), (68), (89), (90), (108). In his Epistle (page 2), Oughtred characterizes the make up of the book in the following terms:

In reading it . . . I met with such a patchery and confusion of
disjoynted stuffe, that I was striken with a new wonder, that any man
should be so simple, as to shame himselfe to the world with such a
hotch-potch.

Grammelogia V differs from Grammelogia IV in having only the second title-page. The first title-page may have been torn off from the copy I have seen. A second difference is that the page with the printed numeral 22 in Grammelogia IV has after the word “Finis” the following notice:

This instrument is made in Silver, or Brasse for the Pocket, or at any
other bignesse, over against Saint Clements Church without Temple Barre,
by Elias Allen.

This notice occurs also on page 22 of Grammelogia I and III, but is omitted from page 22 of Grammelogia V.

Description of Delamain’s instrument of 1630

In his address to King Charles I, in his Grammelogia I, Delamain emphasizes the ease of operating with his slide rule by stating that it is “fit for use . . . as well on Horse backe as on Foot.” Speaking “To the Reader,” he states that he has “for many yeares taught the Mathematicks in this Towne,” and made efforts to improve Gunter’s scale “by some Motion, so that the whole body of Logarithmes might move proportionally the one to the other, as occasion required. This conceit in February last [1629] I struke upon, and so composed my Grammelogia or Mathematicall Ring; by which only with an ocular inspection, there is had at one instant all proportionalls through the said body of Numbers.” He dates his preface “first of January, 1630.” The fifth and sixth pages contain his “Description of the Grammelogia,” the term Grammelogia being applied to the instrument, as well as to the book. His description is as follows:

The parts of the Instrument are two Circles, the one moveable, and the
other fixed; The moveable is that unto which is fastened a small pin to
move it by; the other Circle may be conceived to be fixed; The
circumference of the moveable Circle is divided into unequall parts,
charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures
doe represent themselves, or such numbers unto which a Cipher or Ciphers
are added, and are varied as the occasion falls out in the speech of
Numbers, so 1. stands for 1. or 10. or 100., &c. the 2. stands for 2. or
20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000., &c.

After elaborating this last point and explaining the decimal subdivisions on the scales of the movable circle, he says that “the numbers and divisions on the fixed Circle, are the very same that the moveable are, . .” There is no drawing of the slide rule in this publication. The twenty-two numbered pages give explanations of the various uses to which the instrument can be put: “How to performe the Golden Rule” (pp. 1-3), “Further uses of the Golden Rule” (pp. 4-6), “Notions or Principles touching the disposing or ordering of the Numbers in the Golden Rule in their true places upon the Grammelogia” (pp. 7-11), “How to divide one number by another” (pp. 12, 13), “to multiply one Number by another” (pp. 14, 15), “To find Numbers in continuall proportion” (pp. 16, 17), “How to extract the Square Root,” “How to extract the Cubicke Root” (pp. 18-21), “How to performe the Golden Rule” (the rule of proportion) is explained thus:

Seeke the first number in the moveable, and bring it to the second number
in the fixed, so right against the third number in the moveable, is the
answer in the fixed.

If the Interest of 100. li. be 8. li. in the yeare, what is the Interest
of 65. li. for the same time.

Bring 100. in the moveable to 8. in the fixed, so right against 65. in
the moveable is 5.2. in the fixed, and so much is the Interest of 65. li.
for the yeare at 8. li. for 100. li. per annum.

The Instrument not removed, you may at one instant right against any
summe of money in the moveable, see the Interest thereof in the fixed:
the reason of this is from the Definition of Logarithmes.

These are the earliest known printed instructions on the use of a slide rule. It will be noticed that the description of the instrument at the opening makes no references to logarithmic lines for the trigonometric functions; only the line of numbers is given. Yet the title-page promised the “resolution of Plaine and Sphericall Triangles.” Page 22 throws light upon this matter:

If there be composed three Circles of equal thicknesse, A.B.C. so that
the inner edge of D [should be B] and the outward edge of A bee
answerably graduated with Logarithmall signes [sines], and the outward
edge of B and the inner edge of A with Logarithmes; and then on the
backside be graduated the Logarithmall Tangents, and againe the
Logarithmall signes oppositly to the former graduations, it shall be
fitted for the resolution of Plaine and Sphericall Triangles.

After twelve lines of further remarks on this point he adds:

Hence from the forme, I have called it a Ring, and Grammelogia by
annoligie of a Lineary speech; which Ring, if it were projected in the
convex unto two yards Diameter, or thereabouts, and the line Decupled, it
would worke Trigonometrie unto seconds, and give proportionall numbers
unto six places only by an ocular inspection, which would compendiate
Astronomicall calculations, and be sufficient for the Prosthaphaeresis of
the Motions: But of this as God shall give life and ability to health and
time.

The unnumbered page following page 22 contains the patent and copyright on the instrument and book:

Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs
an Instrument called Grammelogia, or The Mathematicall Ring, together
with a Booke so intituled, expressing the use thereof, being his owne
Invention; we of our Gracious and Princely favour have granted unto the
said Richard Delamain and his Assignes, Privilege, Licence, and
Authority, for the sole Making, Printing and Selling of the said
Instrument and Booke: straightly forbidding any other to Make, Imprint,
or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument
or Booke within any our Dominions, during the space of ten yeares next
ensuing the date hereof, upon paine of Our high displeasure. Given under
our hand and Signet at our Palace of Westminster, the fourth day of
January, in the sixth yeare of our Raigne.

Delamain’s later designs, and directions for using his instruments

In the Appendix of Grammelogia III, on page 52 is given a description of an instrument promised near the end of Grammelogia I:

That which I have formerly delivered hath been onely upon one of the
Circles of my Ring, simply concerning Arithmeticall Proportions, I will
by way of Conclusion touch upon some uses of the Circles, of Logarithmall
Sines, and Tangents, which are placed on the edge of both the moveable
and fixed Circles of the Ring in respect of Geometricall Proportions, but
first of the description of these Circles.

First, upon the side that the Circle of Numbers is one, are graduated on
the edge of the moveable, and also on the edge of the fixed the
Logarithmall Sines, for if you bring 1. in the moveable amongst the
Numbers to 1. in the fixed, you may on the other edge of the moveable and
fixed see the sines noted thus 90. 90. 80. 80. 70. 70. 60. 60. &c. unto
6.6. and each degree subdivided, and then over the former divisions and
figures 90. 90. 80. 80. 70. 70. &c. you have the other degrees, viz. 5.
4. 3. 2. 1. each of those divided by small points.

Secondly, (if the Ring is great) neere the outward edge of this side of
the fixed against the Numbers, are the usuall divisions of a Circle, and
the points of the Compasse: serving for observation in Astronomy, or
Geometry, and the sights belonging to those divisions, may be placed on
the moveable Circle.

Thirdly, opposite to those Sines on the other side are the Logarithmall
Tangents, noted alike both in the moveable and fixed thus
6.6.7.7.8.8.9.9.10.10.15.15.20.20. &c. unto 45.45. which numbers or
divisions serve also for their Complements to 90. so 40 gr. stands for
50. gr. 30. gr. for 60 gr. 20. gr. for 70. gr. &c. each degree here both
in the moveable and fixed is also divided into parts. As for the degrees
which are under 6. viz. 5.4.3.2.1. they are noted with small figures over
this divided Circle from 45.40.35.30.25. &c. and each of those degrees
divided into parts by small points both in the moveable and fixed.

Fourthly, on the other edge of the moveable on the same side is another
graduation of Tangents, like that formerly described. And opposite unto
it, in the fixed is a Graduation of Logarithmall sines in every thing
answerable to the first descrition of Sines on the other side.

Fifthly, on the edge of the Ring is graduated a parte of the Æquator,
numbered thus 10 20. 30. unto 100. and there unto is adjoyned the degrees
of the Meridian inlarged, and numbered thus 10 20.30 unto 70. each degree
both of the Æquator, and Meridian are subdivided into parts; these two
graduated Circles serve to resolve such Questions which concerne
Latitude, Longitude, Rumb, and Distance, in Nauticall operations.

Sixthly, to the concave of the Ring may be added a Circle to be elevated
or depressed for any Latitude, representing the Æquator, and so divided
into houres and parts with an Axis, to shew both the houre, and Azimuth,
and within this Circle may be hanged a Box, and Needle with a Socket for
a staffe to slide into it, and this accommodated with scrue pines to
fasten it to the Ring and staffe, or to take it off at pleasure.

The pages bearing the printed numbers 53-68 in the Grammelogia III, IV and V make no reference to the dispute with Oughtred and may, therefore, be assumed to have been published before the appearance of Oughtred’s Circles of Proportion. On page 53, “To the Reader,” he says:

. . . you may make use of the Projection of the Circles of the Ring upon
a Plaine, having the feet of a paire of compasses (but so that they be
flat) to move on the Center of that Plaine, and those feet to open and
shut as a paire of Compasses . . . now if the feet bee opened to any two
termes or numbers in that Projection, then may you move the first foot to
the third number, and the other foot shall give the Answer; . . . it hath
pleased some to make use of this way. But in this there is a double
labour in respect to that of the Ring, the one in fitting those feet unto
the numbers assigned, and the other by moving them about, in which a man
can hardly accommodate the Instrument with one hand, and expresse the
Proportionals in writing with the other. By the Ring you need not but
bring one number to another, and right against any other number is the
Answer without any such motion. . . . upon that [the Ring] I write,
shewing some uses of those Circles amongst themselves, and conjoyned with
others . . . in Astronomy, Horolographie, in plaine Triangles applyed to
Dimensions, Navigation, Fortification, etc. . . . But before I come to
Construction, I have thought it convenient by way introduction, to
examine the truth of the graduation of those Circles . . .

These are the words of a practical man, interested in the mechanical development of his instrument. He considers not only questions of convenience but also of accuracy. The instrument has, or may have now, also lines of sines and tangents. To test the accuracy of the circles of Numbers, “bring any number in the moveable to halfe of that number in the fixed: so any number or part in the fixed shall give his double in the moveable, and so may you trie of the thirds, fourths &c. of numbers, vel contra,” (p. 54). On page 55 are given two small drawings, labelled, “A Type of the Ringe and Scheme of this Logarithmicall projection, the use followeth. These Instruments are made in Silver or Brasse by John Allen neare the Sauoy in the Strand.”

IV. CONTROVERSY BETWEEN OUGHTRED AND DELAMAIN ON THE INVENTION OF THE
CIRCULAR SLIDE RULE

Delamain’s publication of 1630 on the ‘Mathematicall Ring’ does not appear at that time to have caused a rupture between him and Oughtred. When in 1631 Delamain brought out his Horizontall Quadrant, the invention of which Delamain was afterwards charged to have stolen from Oughtred, Delamain was still in close touch with Oughtred and was sending Oughtred in the Arundell House, London, the sheets as they were printed. Oughtred’s reference to this in his Epistle (p. 20) written after the friendship was broken, is as follows:

While he was printing his tractate of the Horizontall quadrant, although
he could not but know that it was injurious to me in respect of my free
gift to Master Allen, and of William Forster, whose translation of my
rules was then about to come forth: yet such was my good nature, and his
shamelessnesse, that every day, as any sheet was printed, hee sent, or
brought the same to mee at my chamber in Arundell house to peruse which I
lovingly and ingenuously did, and gave him my judgment of it.

Even after Forster’s publication of Oughtred’s Circles of Proportion, 1632, Oughtred had a book, A canon of Sines Tangents and Secants, which he had borrowed from Delamain and was then returning (Epistle, page (5)). The attacks which Forster, in the preface to the Circles of Proportion, made upon Delamain (though not naming Delamain) started the quarrel. Except for Forster and other pupils of Oughtred who urged him on to castigate Delamain, the controversy might never have arisen. Forster expressed himself in part as follows:

. . . being in the time of the long vacation 1630, in the Country, at the
house of the Reverend, and my most worthy friend, and Teacher, Mr.
William Oughtred (to whose instruction I owe both my initiation, and
whole progresse in these Sciences.) I vpon occasion of speech told him of
a Ruler of Numbers, Sines, & Tangents, which one had be-spoken to be made
(such as it vsually called Mr. Gunter’s Ruler) 6 feet long, to be vsed
with a payre of beame-compasses. “He answered that was a poore invention,
and the performance very troublesome: But, said he, seeing you are taken
with such mechanicall wayes of Instruments, I will shew you what deuises
I have had by mee these many yeares.” And first, hee brought to mee two
Rulers of that sort, to be vsed by applying one to the other, without any
compasses: and after that hee shewed mee those lines cast into a circle
or Ring, with another moueable circle vpon it. I seeing the great
expeditenesse of both those wayes; but especially, of the latter, wherein
it farre excelleth any other Instrument which hath bin knowne; told him,
I wondered that hee could so many yeares conceale such vseful inuentions,
not onely from the world, but from my selfe, to whom in other parts and
mysteries of Art, he had bin so liberall. He answered, “That the true way
of Art is not by Instruments, but by Demonstration: and that it is a
preposterous course of vulgar Teachers, to begin with Instruments, and
not with the Sciences, and so in-stead of Artists, to make their
Schollers only doers of tricks, and as it were Iuglers: to the despite of
Art, losse of precious time, and betraying of willing and industrious
wits, vnto ignorance and idlenesse. That the vse of Instruments is indeed
excellent, if a man be an Artist: but contemptible, being set and opposed
to Art. And lastly, that he meant to commend to me, the skill of
Instruments, but first he would haue me well instructed in the Sciences.
He also shewed me many notes, and Rules for the vse of those circles, and
of his Horizontall Instrument, (which he had proiected about 30 yeares
before) the most part written in Latine. All which I obtained of him
leaue to translate into English, and make publique, for the vse, and
benefit of such as were studious, and louers of these excellent Sciences.

Which thing while I with mature, and diligent care (as my occasions would
give me leaue) went about to doe: another to whom the Author in a louing
confidence discouered this intent, using more hast then good speed, went
about to preocupate; of which vntimely birth, and preuenting (if not
circumuenting) forwardnesse, I say no more: but aduise the studious
Reader, onely so farre to trust, as he shal be sure doth agree to truth &
Art.

While in this dedication reference is made to a slide rule or “ring” with a “moveable circle,” the instrument actually described in the Circles of Proportion consists of fixed circles “with an index to be opened after the manner of a paire of Compasses.” Delamain, as we have seen, had decided preference for the moveable circle. To Oughtred, on the other hand, one design was about as good as the other; he was more of a theorist and repeatedly expressed his contempt for mathematical instruments. In his Epistle (page (25)), he says he had not “the one halfe of my intentions upon it” (the rule in his book), nor one with a “moveable circle and a thread, but with an opening Index at the centre (if so be that bee cause enough to make it to bee not the same, but another Instrument) for my part I disclaime it: it may go seeke another Master: which for ought I know, will prove to be Elias Allen himselfe: for at his request only I altered a little my rules from the use of the moveable circle and the thread, to the two armes of an Index.”

All parts of Delamain’s Grammelogia IV, except pages 1-22 and 53-68 considered above, were published after the Circles of Proportion, for they contain references to the ill treatment that Delamain felt or made believe that he felt, that he had received in the book published by Oughtred and Forster. Oughtred’s reference to teachers whose scholars are “doers of tricks,” “Iuglers,” and Forster’s allusion to “another to whom the Author in a loving confidence” explained the instrument and who “went about to preocupate” it, are repeatedly mentioned. Delamain says, (page (89)) that at first he did not intend to express himself in print, “but sought peace and my right by a private and friendly way.” Oughtred’s account of Delamain’s course is that of an “ill-natured man” with a “virulent tongue,” “sardonical laughter” and “malapert sawsiness.” Contrasting Forster and Delamain, he says that, of the former he “had the very first moulding” and made him feel that “the way of Art” is “by demonstration.” But Delamain was “already corrupted with doing upon Instruments, and quite lost from ever being made an Artist.” (Epistle page (27)). Repeatedly does Oughtred assert Delamain’s ignorance of mathematics. The two men were evidently of wholly different intellectual predilections. That Delamain loved instruments is quite evident, and we proceed to describe his efforts to improve the circular slide rule.

The Grammelogia IV is dedicated to King Charles I. Delamain says:

. . . Everything hath his beginning, and curious Arts seldome come to the
height at the first; It was my promise then to enlarge the invention by a
way of decuplating the Circles, which I now present unto your sacred
Majestie as the quintessence and excellencie there of . . .

His enlarged circular rules are illustrated in the Bodleian Library copy of Grammelogia IV by four diagrams, two of them being the two drawings on the two title-pages at the beginning of the Grammelogia IV, 4 inches in external diameter, and exhibiting eleven concentric circular lines carrying graduations of different sorts. In the second of these designs all circles are fixed. The other two drawings are each 10¾ inches in external diameter and exhibit 18 concentric circular lines; the folded sheet of the first of these drawings is inserted between pages (23) and (24), the second folded sheet between pages (83) and (84). All circles of this second instrument are fixed. Counting in the two small drawings in Grammelogia III, there are in all six drawings of slide rules in the Bodleian Grammelogia IV. On pages (24) to (43) Delamain explains the graduation of slide rules. He takes first a rule which has one circle of equal parts, divided into 1000 equal divisions. From a table of logarithms he gets log 2 = 0.301; from the number 301 in the circle of equal parts he draws a line to the center of the circle and marks the intersection with the circles of numbers by the figure 2. Thus he proceeds with log 3, log 4, and so on; also with log sin x and log tan x. For log sin x he uses two circles, the first (see page (27)) for angles from 34′ 24″ to 5° 44′ 22″, the second circle from 5° 44′ 22″ to 90°. The drawings do not show the seconds. He suggests many different designs of rules. On page (29) he says:

For the single projection of the Circles of my Ring, and the dividing and
graduating of them: which may bee so inserted upon the edges of Circles
of mettle turned in the forme of a Ring, so that one Circle may moove
betweene two fixed, by helpe of two stayes, then may there be graduated
on the face of the Ring, upon the outer edge of the mooveable and inner
edge of the fixed, the Circle of Numbers, then upon the inner edge of
that mooveable Circle, and the outward edge of that inner fixed Circle
may be inserted the Circle of Sines, and so according to the description
of those that are usually made.

In addition to these lines he proceeds to mention the circle giving the ordinary division into degrees and minutes, and two circles of tangents on the other side of the rule.

Next Delamain explains an arrangement of all the graduation on one side of the rule by means of “a small channell in the innermost fixed Circle, in which may be placed a small single Index, which may have sufficient length to reach from the innermost edge of the Mooveable Circle, unto the outmost edge of the fixed Circle, which may be mooved to and fro at pleasure, in the channell, which Index may serve to shew the opposition of Numbers” (p. (31)). From this it is clear that the invention of the “runner” goes back to the very first writers on the slide rule.

After describing a modification of the above arrangement, he adds, “many other formes might be deliverd, about this single projection” (p. (32)).

Proceeding to the “enlarging” of the circles in the Ring, to, say, the “Quadruple to that which is single, that is, foure times greater,” the “equall parts” are distributed over four circles instead of only one circle, but the general method of graduation is the same as before (p. (33)); there being now four circles carrying the logarithms of numbers, and so on. Next he points out “severall wayes how the Circles of the Mathematicall Ring (being inlarged) may be accommodated for practicall use:” (1) The Circles are all fixed in a plain and movable flat compasses (or better, a movable semicircle) are used for fixing any two positions; (2) There is a “double projection” of each logarithmic line “inlarged on a Plaine,” one fixed, the other movable, as shown in his first figure on the title-page, a single index only being used; (3) use of “my great Cylinder which I have long proposed (in which all the Circles are of equall greatnesse,) and it may be made of any magnitude or capacity, but for a study (hee that will be at the charge) it may be of a yard diameter and of such an indifferent length that it may containe 100 or more Circles fixed parallel one to the other on the Cylinder, having a space betweene each of them, so that there may bee as many mooveable Circles, as there are fixed ones, and these of the mooveable linked, or fastened together, so that they may all moove together by the fixed ones in these spaces, whose edges both of the fixed, and mooveable being graduated by helpe of a single Index will shew the proportionalls by opposition in this double Projection, or by a double Index in a single Projection” (p. (36)).

Next follows the detailed description of his Ring “on a Plaine, according to the diagramme that was given the King (for a view of that projection) and afterwards the Ring it selve.” The diagram is the large one which we mentioned as inserted between pages (23) and (24). The instrument has two circles, one moveable, upon each of which are described 13 distinct circular graduations. The lines on the fixed circle are: “The Circle of degrees and calendar,” E. “Circle of equall parts, and part of the Equator, and Meridian,” TT. “The Circle of Tangents,” S. “The Circle of Sines,” D. “The Circle of Decimals,” N. “The Circle of Numbers.” The lines on the movable circle are: N. “The Circle of Numbers,” E. “The Circle of equated figures, and bodies,” S. “The Circle of Sines,” TT. “The Circle of Tangents,” Y. “The Circle of time, yeares, and monethes.”

On pages (84)-(88) Delamain explains an enlargement of his Ring for computations involving the sines of angles near to 90°. On page (86) he says:

I have continued the Sines of the Projection unto two severall
revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr.
(being the last revolution of the decuplation of the former, or the
hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s.
and ends at 90.gr. (being the last of a ternary of decuplated
revolutions, or the thousand part of that Projection) and may bee thus
used.

He explains the manner of using these extra graduations. Thus he claims to have attained degrees of accuracy which enabled him to do what “some one” had declared “could not bee done.” It is hardly necessary to point out that Delamain’s Grammelogia IV suggests designs of slide rules which inventors two hundred or more years later were endeavouring to produce. Which of Delamain’s designs of rules were actually made and used, he does not state explicitly. He refers to a rule 18 inches in diameter as if it had been actually constructed (pages (86), (88)). Oughtred showed no appreciation of such study in designing and ridiculed Delamain’s efforts, in his Epistle.

Additional elucidations of his designs of rules, along with explanations of the relations of his work to that of Gunter and Napier, and sallies directed against Oughtred and Forster, are contained on pages (8)-(21) of his Grammelogia IV.

V. INDEPENDENCE AND PRIORITY OF INVENTION

The question of independence and priority of invention is discussed by Delamain more specifically on pages (89)-(113); Oughtred devotes his entire Epistle to it. It is difficult to determine definitely which publication is the later, Delamain’s Grammelogia IV or Oughtred’s Epistle. Each seems to quote from the other. Probably the explanation is that the two publications contain arguments which were previously passed from one antagonist to the other by word of mouth or by private letter. Oughtred refers in his Epistle (p. (12)) to a letter from Delamain. We believe that the Epistle came after Delamain’s Grammelogia IV. Delamain claims for himself the invention of the circular slide rule. He says in his Grammelogia IV. (p. (99)), “when I had a sight of it, which was in February, 1629 (as I specified in my Epistle) I could not conceale it longer, envying my selfe, that others did not tast of that which I found to carry with it so delightfull and pleasant a goate [taste] . . .” Delamain asserts (without proof) that Oughtred “never saw it as he now challengeth it to be his invention, untill it was so fitted to his hand, and that he made all his practise on it after the publishing of my Booke upon my Ring, and not before; so it was easie for him or some other to write some uses of it in Latin after Christmas, 1630 and not the Sommer before, as is falsely alledged by some one . . .” (p. (91)). Delamain’s accusation of theft on the part of Oughtred cannot be seriously considered. Oughtred’s reputation as a mathematician and his standing in his community go against such a supposition. Moreover, William Forster is a witness for Oughtred. The fact that Oughtred had the mastery of the rectilinear slide rule as well, while Delamain in 1630 speaks only of the circular rule, weighs in Oughtred’s favour.

Oughtred says he invented the slide rule “above twelve yeares agoe,” that is, about 1621, and “I with mine owne hand made me two such Circles, which I have used ever since, as my occasions required,” (Epistle p. (22)). On the same page, he describes his mode of discovery thus:

I found that it required many times too great a paire of Compasses [in
using Gunter’s line], which would bee hard to open, apt to slip, and
troublesome for use. I therefore first devised to have another Ruler with
the former: and so by setting and applying one to the other, I did not
onely take away the use of Compasses, but also make the worke much more
easy and expedite: when I should not at all need the motion of my hand,
but onely the glancing at my sight: and with one position of the Rulers,
and view of mine eye, see not one onely, but the manifold proportions
incident unto the question intended. But yet this facility also wanted
not some difficulty especially in the line of tangents, when one arch was
in the former mediety of the quadrant, and the other in the latter: for
in this case it was needful that either one Ruler must bee as long againe
as the other; or else that I must use an inversion of the Ruler, and
regression. By this consideration I first of all saw that if those lines
upon both Rulers were inflected into two circles, that of the tangents
being in both doubled, and that those two Circles should move one upon
another; they with a small thread in the center to direct the sight,
would bee sufficient with incredible and wonderfull facility to worke all
questions of Trigonometry . . .

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)