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.”