Florian Cajori

“ 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. ”

“ 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. ”

“ 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 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. ”

“ 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. ”

“ On the History of Gunter’s Scale and the Slide Rule during the Seventeenth Century, by Florian Cajori. ”

“ 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. ”

“ 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. ”

“ At one time Delamain, a teacher of mathematics in London, was assisted by Oughtred in his mathematical studies. In 1630 Delamain published the Grammelogia, a pamphlet describing a circular slide rule and its use. In 1631 he published another tract, on the Horizontall Quadrant. [16] In 1632 appeared Oughtred’s Circles of Proportion [17] translated into English from Oughtred’s Latin manuscript by another pupil, William Forster, in the preface of which Forster makes the charge (without naming Delamain) that “another . . . went about to pre-ocupate” the new invention. ”

“ 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. ”

“ 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 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. ”

“ 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. ”

“ 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 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 ”

“ He gives no drawing of his “triangular quadrant,” hence his account of it is unsatisfactory. He explains the use of “gage-points.” His placing logarithmic lines on the edges of instrument boxes was outdone in oddity later by Everard who placed them on tobacco-boxes. [34] 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. ”

“ 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.” ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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:’
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 . ”

“ 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 ”

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MATHEMATICS.—Mellen W. Haskell, Editor. Price per volume, $5.00. ”