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Title: On the History of Gunter's Scale and the Slide Rule During the
Seventeenth Century

Author: Florian Cajori

Release date: February 26, 2013 [eBook #42216]

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK ON THE
HISTORY OF GUNTER'S SCALE AND THE SLIDE RULE DURING

THE SEVENTEENTH CENTURY ***

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

[187

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

[1] [2]
In my history of the slide rule , and my article on its invention 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
[3]
invention to himself and as having written a scurrilous pamphlet against
Oughtred. All our information about Delamain was taken from De [188
[4]
Morgan, 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
[5]
Gunter’s scale into France.

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
[6]
Gunter himself had learnedly explained its use in a far larger volume.

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 [189
devoid of interest. The editions of 1645 and 1658 contain an important

[7]
innovation. 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 [190
antiquarian, Elias Ashmole, on August 13 [1646?]; “I bought of Mr.
[8] [9]
Milbourn all his Books and Mathematical Instruments.” Charles Hutton
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
[10]
William Oughtred in his Epistle 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
[11]
revolution being but two true semicircles described on severall centers.”

Changes introduced by Thomas Brown and John Brown

Thomas Brown did not publish any description of his instrument, but his son,
[12]
John Brown, published in 1661 a small book, 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 [191
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
[13] [192
later publications, 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
[14]
Partridge’s slide rule, returns to forms of Gunter’s scale, saying:

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.
[15]
James Atkinson 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
[16]
its use. In 1631 he published another tract, on the Horizontall Quadrant.
[17]
In 1632 appeared Oughtred’s Circles of Proportion 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.”

[194

Different editions or impressions

There are at least five different editions, or impressions, of the Grammelogia
which we designate, for convenience, as follows:
[18]
Grammelogia I, 1630. One copy in the Cambridge University Library.

Grammelogia II, I have not seen a copy of this.
[19]
Grammelogia III, One copy in the Cambridge University Library.

Grammelogia IV, One copy in the British Museum, another in the
[20]
Bodleian Library, Oxford.

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. [195
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.

[196

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 [197
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 [198
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). [199
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 [200
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: [201

. . . 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 [202
“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 [203
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 [204
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 Oughtred’s Statement
Grammelogia IV, page (98) Epistle, page (23)

“. . . about Alhalontide 1630. (as “Shortly after my gift to Elias
our Authors reporteth) was the time Allen, I chanced to meet with
he was circumvented, and then his Richard Delamain in the street (it

intent in a loving manner (as was at Alhallontide) and as we
before) he opened unto me, which walked together I told him what an
particularly I will dismantle in the Instrument I had given to Master
very naked truth: for, wee being Allen, both of the Logarithmes
walking together some few weekes projected into circles, which being
before Christmas, upon Fishstreet lesse then one foot diameter would
hill, we discoursed upon sundry performe as much as one of Master
things Mathematicall, both Gunters Rulers of sixe feet long:
Theoreticall and Practicall, and of and also of the Prostaphaereses of
the excellent inventions and helpes the Plannets and second motions.
that in these dayes were produced, Such an invention have I said he:
amongst which I was not a little for now his intentions (that is his
taken with that of the Logarythmes, ambition) beganne to worke: . . .
commending greatly the ingenuitie But he saith, Then after my
of Mr. Gunter in the Projection, and comming home I sent him a sight of
inventing of his Ruler, in the lines my projection drawne in past-
of proportion, extracted from these board. See how notoriously he
Logarythmes for ordinary jugleth without an Instrument. Then
Practicall uses; He replyed unto me after: how long after? a sight of my
(in these very words) What will yov projection: of how much? More
say to an Invention that I have, then seven weekes after on
which in a lesse extent of the December 23, he sent to mee the
Compasses shall worke truer then line of numbers onely set upon a
that of Mr. Gunters Ruler, I asked circle: . . . and so much onely he
him then of what forme it was, he presented to his Majesty: but as for
answered with some pause (which Sine or tangent of his, there was not
no doubt argued his suspition of the least shew of any. Neither could
mee that I might conceive it) that it he give to Master Allen any
was Arching-wise, but now hee direction for the composure of the
sayes that hee told mee then, it was circles of his Ring, or for the
Circular (but were I put to my oath division of them: as upon his oath
to avoid the guilt of Conscience I Master Allen will testify how hee
would conclude in the former.) At misled him, and made him labour in
which immediately I answered, I vain above three weeks together,
had the like my selfe, and so we until Master Allen himselfe found
discoursed not a word more out his ignorance and mistaking,
touching that subject . . . Then after which is more cleare then is
my coming home I sent him a sight

of my Projection drawne in possible with any impudence to be
Pastboard: Now admit I had not the outfaced.”
Invention of my Ring before I
discoursed . . . it was not so facil [205
for mee . . . to raise and compose so
complete, and absolute an
Instrument from so small a
principle, or glimpse of light . . .”

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.
[21]
Doubtless few people read both sides. Oughtred says: “this scandall . . .
hath with them, to whom I am not knowne, wrought me much prejudice and
[22]
disadvantage . .” Aubrey, 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
[23]
his own labour . . .

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 [206
Delamain was appointed tutor to the King in mathematics. His widow
[24]
petitioned the House of Lords in 1645 for relief; he had ten children.

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
[25]
wrought by it in the mathematics.”

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
[27]
Oughtred to design a gauging rod. 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
[28]
Oughtred, wrote him as follows:

I have light upon your little book of artificial gauging, wherewith I [207
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
[29]
11, 1635 thus:

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.
[30]
On April 20, 1638, we find Oughtred giving Elias Allen directions “about
[31]
the making of the two rulers.” As in 1633, 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,
[32]
Henry Coggeshall, W. Hunt and Sir Isaac Newton. 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 [208
it is unsatisfactory. He explains the use of “gage-points.” His placing
logarithmic lines on the edges of instrument boxes was outdone in oddity
[34]
later by Everard who placed them on tobacco-boxes. 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.
[36]
Later further changes were introduced in Coggeshall’s rule.

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
[37]
Leupold’s description 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, [209
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
[38]
we have mentioned earlier. He says: “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
[39]
of the rectilinear type.

January 12, 1915.

Footnotes

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

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

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

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

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

[6]
The full 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:

Gram̄ elogia | 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. |

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.

[21]
Epistle, p. (8).

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

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

[24]
Dictionary 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.

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

[26]
The 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.

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

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

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

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

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

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

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

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

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

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

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

[39]
Probably 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.]
[211

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Volume III. Business Cycles, by Wesley C. Mitchell. 1913
$5.00; cloth, 7.50

Volume IV. 1. The Battle of the Seven Arts, by Louis J. Paetow. July,
1914
.75

UNIVERSITY OF CALIFORNIA CHRONICLE.—An official record of
University life, issued quarterly. George M. Calhoun, Editor. Price,
$1.00 per year. Current, Volume XXII.

Transcriber’s Notes

Inserted subchapter headings from the Table of Contents into the text.
Corrected two page numbers in the Table of Contents
Corrected a Greek quotation in footnote 6 by reference to the original
sources.

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