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.