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-compa.s.ses. "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 compa.s.ses: 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 circ.u.muenting) 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 Compa.s.ses." 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 a.s.sert 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 ill.u.s.trated in the Bodleian Library copy of Grammelogia IV by four diagrams, two of them being the two drawings on the two t.i.tle-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 10 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 compa.s.ses (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 t.i.tle-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 s.p.a.ce 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 s.p.a.ces, whose edges both of the fixed, and mooveable being graduated by helpe of a single Index will shew the proportionalls by opposition in this double Projection, or by a double Index in a single Projection" (p. (36)).

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

"The Circle of Decimals," N. "The Circle of Numbers." The lines on the movable circle are: N. "The Circle of Numbers," E. "The Circle of equated figures, and bodies," S. "The Circle of Sines," TT. "The Circle of Tangents," Y. "The Circle of time, yeares, and monethes."

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

I have continued the Sines of the Projection unto two severall revolutions, the one beginning at 77.gr. 45.m. 6.s. and ends at 90.gr.

(being the last revolution of the decuplation of the former, or the hundred part of that Projection) the other beginning at 86.gr. 6.m. 48.s.

and ends at 90.gr. (being the last of a ternary of decuplated revolutions, or the thousand part of that Projection) and may bee thus used.

He explains the manner of using these extra graduations. Thus he claims to have attained degrees of accuracy which enabled him to do what "some one"

had declared "could not bee done." It is hardly necessary to point out that Delamain's Grammelogia IV suggests designs of slide rules which inventors two hundred or more years later were endeavouring to produce. Which of Delamain's designs of rules were actually made and used, he does not state explicitly. He refers to a rule 18 inches in diameter as if it had been actually constructed (pages (86), (88)). Oughtred showed no appreciation of such study in designing and ridiculed Delamain's efforts, in his Epistle.

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

V. INDEPENDENCE AND PRIORITY OF INVENTION

The question of independence and priority of invention is discussed by Delamain more specifically on pages (89)-(113); Oughtred devotes his entire Epistle to it. It is difficult to determine definitely which publication is the later, Delamain's Grammelogia IV or Oughtred's Epistle. Each seems to quote from the other. Probably the explanation is that the two publications contain arguments which were previously pa.s.sed 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 a.s.serts (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 Compa.s.ses [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 Compa.s.ses, but also make the worke much more easy and expedite: when I should not at all need the motion of my hand, but onely the glancing at my sight: and with one position of the Rulers, and view of mine eye, see not one onely, but the manifold proportions incident unto the question intended. But yet this facility also wanted not some difficulty especially in the line of tangents, when one arch was in the former mediety of the quadrant, and the other in the latter: for in this case it was needful that either one Ruler must bee as long againe as the other; or else that I must use an inversion of the Ruler, and regression. By this consideration I first of all saw that if those lines upon both Rulers were inflected into two circles, that of the tangents being in both doubled, and that those two Circles should move one upon another; they with a small thread in the center to direct the sight, would bee sufficient with incredible and wonderfull facility to worke all questions of Trigonometry ...

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

Delamain's Statement Grammelogia IV, page (98)

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

Oughtred's Statement Epistle, page (23)

"Shortly after my gift to Elias Allen, I chanced to meet with Richard Delamain in the street (it was at Alhallontide) and as we walked together I told him what an Instrument I had given to Master Allen, both of the Logarithmes projected into circles, which being lesse then one foot diameter would performe as much as one of Master Gunters Rulers of sixe feet long: and also of the Prostaphaereses of the Plannets and second motions. Such an invention have I said he: for now his intentions (that is his ambition) beganne to worke: ... But he saith, Then after my comming home I sent him a sight of my projection drawne in past-board.

See how notoriously he jugleth without an Instrument. Then after: how long after? a sight of my projection: of how much? More then seven weekes after on December 23, he sent to mee the line of numbers onely set upon a circle: ... and so much onely he presented to his Majesty: but as for Sine or tangent of his, there was not the least shew of any. Neither could he give to Master Allen any direction for the composure of the circles of his Ring, or for the division of them: as upon his oath Master Allen will testify how hee misled him, and made him labour in vain above three weeks together, until Master Allen himselfe found out his ignorance and mistaking, which is more cleare then is possible with any impudence to be outfaced."

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

Delamain hearing that Brown with his Serpentine had another line by which he could worke to minutes in the 90 degree of sines ... gave the [his]

booke to Browne: who in thankfulnesse could not but gratify Delamain with his Lines also: and teach him the use of them, but especially of the great Line: with this caution on both sides, that one should not meddle with the others invention. Two dayes after Delamain ... because he had found some things to be altered therin, ... asked for the booke ...

but as soone as he had got it in his hands he rent out all the middle part with the two Schemes & put them up in his pocket & went his way . .

. and ... laboureth to recall all the bookes he had given forth ...

And shortly after this he got a new Printer (who was ignorant of his former Schemes) to print him new: giving him an especiall charge of the outermost line newly graven in the Plate, which indeed is Brownes very line: and then altering his book ...

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

The effect of this controversy upon interested friends was probably small.

Doubtless few people read both sides. Oughtred says:[21] "this scandall . .

. hath with them, to whom I am not knowne, wrought me much prejudice and disadvantage . ." Aubrey,[22] a friend of Oughtred, refers to Delamain "who was so sawcy to write against him" and remembers having seen "many yeares since, twenty or more good verses made" against Delamain. Another friend of Oughtred, William Robinson, who had seen some of Delamain's publications, but not his Grammelogia IV, wrote in a letter to Oughtred, shortly before the appearance of the latter's Epistle:

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

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

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

VI. OUGHTRED'S GAUGING LINE, 1633

It has not been generally known, hitherto, that Oughtred designed a rectilinear slide rule for gauging and published a description thereof in 1633.[26] In his Circles of Proportion, chapter IX, Oughtred had offered a closer approximation than that of Gunter for the capacity of casks. The Gauger of London expostulated with Oughtred for presuming to question anything that Gunter had written. The ensuing discussion led to an invitation extended by the Company of Vintners to the instrument maker Elias Allen to request Oughtred to design a gauging rod.[27] This he did, and Allen received an order for "threescore" instruments. On page 19 Oughtred describes his 'Gauging Rod:'