Oughtred's Circles of Proportion

* Introduction

The following page contains sections of William Oughtred's Circles of Proportion. The circles have an interesting bibliographical history but this is not the place to elaborate on those details. I have not, as yet, been able to include the full text. Any errors and omissions from the original are mine alone. I have tried to preserve the original spelling.

Copyright, Christopher Sangwin 2000

* Contents

Front piece
Epistle Dedicatorie
To the honourable and renowned for vertue, learning, and true valour, Sir Kenelme Digbye Knight.
The First Part of this Booke
Shewing the use of the First side of the instrument, for the working of the Proportions both simple and compound, and the ready and easie reolving of questions both in Arithmetique, Geometrie, and Astronomie, by Calculation.
To the English Gentrie and all others studious of the Mathematicks which shall bee Readers herof.
The just Apologie of Wil: Oughtred against the slanderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematical Ring or Mirifica logarithmorum projectio circularts.

The Circles of Proportion and the Horizontall instrument.
Both invented, and the uses of both written in Latine by Mr. W. O.
Translated into English and set forth for the publique benefit by William Forster.

Printed for Elias Allen maker of these and all other Methematical Instruments, and are to be sold at his shop over against St Clements Church with out Temple-barr.

The Circles of Proportion and the Horizontall instrument.
Both invented, and the uses of both written in Latine by the
Learned Mathematician Mr. W. O.
Translated into English, and set forth for the publique benefit by
William Forster, lover and practizer of the Mathematical Sciences.


Printed by Avg. Mathewes, dwelling in the Parsonage court, neere St Brides. 1632

Epistle Dedicatorie

To the honourable and renowned for vertue, learning, and true valour, Sir Kenelme Digbye Knight.


The excellent accomplishments wherewith you are adorned both of virtue, and learning, and particularly in the Mathematical Sciences together with the Honourable respect the Author hereof beareth unto your Worth, and his desire to present unto you, and under the happy auspice of your renowned name, to publish to the world this Treatise: the owning where of though I may not chalenge to my selfe, yet the birth and production, whereby it hath a being to the benefit of others, is, as unto a second parent due unto me.

For 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 upon occasion of speech told him of a Ruler of Numbers, Sines, & Tangents, which one had bespoken to be mase (such as in usually called Mr Gunters Ruler) 6 feet long, to be used 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 ways of instruments, I will shew you what devises I had had by mee these many years. And first hee brought to mee two Rulers of that sort, to be used by applying one to the other, without any compasses: and after that hee showed mee those lines cast into a circle of Ring, with another moveable circle upon it, I seeing the great expeditensse of both those wayes, but especially, of the latter, wherein it farre excelleth any other Instrument which hath bin knowne, told him, I woundered that hee could so many yeares conceale such usefull inventions, not only from the World, but from myelf, 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 Artists, to make their Schollers only doers of tricks, as it were Juglers: to the despite of Art, losse of precious time, and betraying of willing and industrious wits, unto ignorance, and idlenesse. That the use 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 have me well instructed in the Sciences. He also showed me many notes, and Rules for the use of those circles, and of his Horizontall Instrument, (which he had projected about 30 years before) the most part written in Latine. All which I obtained of him leave to translate into English, and make publique, for the use, and benefit of such as were studious, & lovers of these excellent Sciences.

Which thing while I with mature, and diligent care (as my occasions would give me leave) went about to doe: another to whom the Author in a loving confidence discoverd this intent, using more hast than good speed, went about to preocupate, of which untimely birth, and preventing (if not circumventing) forwardnesse, I say no more: but advise the Studious Reader, onely so farre to trust, as he shall be sure doth agree to true & Art.

And thus most noble Sir, without and braving flourishes, or needless multiplying of tautologized and erroneous praecepts, in naked truth, and in the modest Simplicity, of the Author himself (whose knowne skill in the whole systeme of Mathematicall learning, having the way made for him, and the subject unvailed, to help his fight), I have not withstanding under the protection of your courteous favour, and learned judgement, persisted in my long conceived purpose of presenting this tractate to the publique view and light. Wishing withall unto your encrease of deserved honor, and happieness

May the 1. 1632
By the honourer
and admirer
of your Worthines
William Forster.


First Part

of this Booke

Shewing the use of the First side of the instrument, for the working of the Proportions both simple and compound, and the ready and easie reolving of questions both in Arithmetique, Geometrie, and Astronomie, by Calculation.

Chap. I.

of the description, and use of the circles in this First Side.

1. There are two sides to the instrument on the one side, as it were in the plaine of the Horizon, is delineated the projection of the Sphaere. On the other side there are diverse kindes of circles, divided after many severall waies, together with an Index to the opened after the manner of a paire of compasses. And of this side we will speake on the first place.

2. The first or outermost circle is of Sines, from 5 degrees 45 minutes almost, untill 90. Each degree till 40 is divided into 12 parts, each part being 5 min: from thence untill 50 deg: into sixe parts which are 10 min: a peece: from thence untill 75 degrees into two parts which are 30 minutes a peece: After that unto 85 deg they are not divided.

3. The second circle is of Tangents, from 5 degrees 45 mins: almost, untill 45 degrees, Every degree being divided into 12 parts which are 5 min: a peece.

4. The Third circle is of Tangents from 45 degrees until 84 degrees 15 mins. Each degree being divided into 12 parts, which are 5 min a peece.

5. The sixt circle is of Tangents from 84 degrees till about 89 degrees 25 minutes.

The Seventh circle is of Tangents from about 35 min till 6 degrees. The Eight circle is of Sines, from about 35 minutes till 6 degrees. [Possible transcription error here]

6. The Fourth circle is of Unaequall Numbers, which are noted with the Figures 2,3,4,5,6,7,8,9,1. Whether you understand them to bee single Numbers, or Tens, Hundreds, or Thousands, &c. And every space of the numbers till 5 is divided into 100 parts, but after 5 till 1, into 50 parts.

The Fourth circle also sheweth the true or naturall Sines, and Tangents. For if the Index bee applyed to any Sine or Tangent, it will cut the true Sine or Tangent in the Fourth Circle. And wee are to knowe that is the Sine or Tangent be in the First, or Second circlem the figures of the Fourth circle doe signifie so many thousands. But if the Sine or Tangent be in the Seventh or Eight circle, the figures in the Fourth circle signifie so many hundreds. And if the Tangents bee ni the Sixt Circle, the figures of the Fourth circle, signifie so many times tenne thousand, or whole Radii.

And by the means the Sine of 23^o,30' will bee found 3987: and the Sine of it's complement 9171. And the Tangent of 23^0,30' will be found 4348: and the Tangent of it's complement, 22998. And the Radius is 10000, that is the figure 1 with four cyphers, or circles. And hereby you may find out both the summe, and also the difference of Sines, and Tangents.

7. The Fith circle is of AEquall numbers, which are noted with the figures 1,2,3,4,5,6,7,8,9,0; and every space is divided into 100 aequall parts.

This Fift circle is scarse of any use, but onely that by helpe there of the given difference of numbers may be multiplied, or divided, as neede shall require.

As for example, if the space between 1.00 and 1.0833+ bee to bee septupled. Apply the Index unto 1.0833+ in the Fourth circle and it will cut in the Fift circle 03476+; which multiplied by 7 makes 24333: then againe, apply the Index unto this number 24333 in the Fift circle, and it will cut in the Fourth circle 1.7512+ And this is the space between 1.00 and 1.0833+ septupled, or the Ratio between 100 and 108 1/3 seven time multiplied into is selfe.

And contrarily, if 1.7512 bee to bee divided by 7: Apply the Index unto 1.7512 in the Fourth circle, and it will be fit circle 24333: which divided by 7 giveth 03476+. Then again unto this Number in the Fift circle apply the rule Index, and in the Fourth circle it wil cut upon 1.0833+ for the septupartion sought for.

The reason of which Operation is, because this Fift circle doth shew Logarithmes of Numbers. For if the index be applyed unto any number in the Fourth circle, it will in the Fiith circle cut upon the Logarithm of the same number, so that to the Logarithme found your prafixe a carateristicall (as Master Briggs terms it) one less then is the number of places of the integers proposed. (which you may rather call the Graduall Number). So the Logarithm of the number 2 will bee found 0.30103 [actually used decimal point here]. And the Logarithme of the Number 43.6 will bee 1.63949 [actually used decimal point here]

Numbers are multiplied by Addition of their Logarithms and they are they are Divided by Subtraction of their Logarithms.

8. In the middest among the Circles is a double Nocturnall instrument, to shew the hower of the night.

9. The right line passing through the center through 90 and 45 I call the Line of Unitie or of the Radiuss.

10. That Arme of the Index which in every operation is placed at the Antecedent, or first terem I call the Antecedent arme: and that which is placed at the consequent terme, I call the consequent Arme.

11. The first significant figure of a number is to be taken in one of those nine figures in the spaces of the fourth circle and the rest of the figures in the divisions and subdivisions following.

But note that for the use of navigation, there have been two circles inserted next within the fift circle which are not in the scheme before pag 1 which makes these innermost circles of the Tangents & Signes to be the eight, nine & tenth.

To the English Gentrie
and all others studious of
the Mathematicks which shall bee
Readers herof.

The just Apologie of Wil: Oughtred against the slanderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematical Ring or Mirifica logarithmorum projectio circularts.

Honourable and much honoured Gentlemen, I was of late at my coming up to London, for the performance of mine ordinary service in the house of my most Honourable Lord the Earle of Arundell and Surry, and Earle Marshall of England, by many of my loving friends presented with a most [...] and scurolous Pamphlet written against me by Richard Delamaine, who professeth himself a Teacher of Mathematickes about the City: Wherein I am brought before you upon the Scaffold, and with all the petulancies of a vexed mind and distempered passion insimulated and charged with, I know not what, injuries (your noble selves also by him ingaged therein, and incensed against me) and at last, as if quite cast, I am schouled by him with a long Lecture or Common, place against Slander and Detraction. I did much wonder at it, to see my self so basely and impudently abused by one whom I never had wronged, but had done very much courteous for giving him acess to my chamber in Arundell House day by day, teaching and instructing him that facultie he professeth: not onely satisfying his scruples in those things he partly knew but even laying the very foundation of diverse parts, whereof hee was utterly ignorant. And I did not so much marvell to see him so bold with me a poore man, but dust and ashes, as I was amazed to see him so fearefully (yet without feare) to play with Amighty God, hypocritically, and aginst his owne conscience in things apparently false, invoking and challenging his all-knowing testimony and in the middest of his mist unmannerly raylings in his booke, and his slaunderous back biting and depraving me, by audacious intruding himself upon my most honourble favours with false complaints, utterly to overthrow and discredite me; in a personated admonition against such uncharitable calumniations, to pronounce judgement against himselfe. I borrowed and perused that worthless Pamphlet, and in reading it (I beshrew him for making me cast away so much of that little time is remayning to my declined years) 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.

In the two first pages (for so he afterwards calleth them) are two Schemes of his Instruments. In the fourth page is his Epistle to the Kings Majestie. In the 5,6,7 are verses to his great commendation. In the 9 me, most plainely still pointing me out, that he needeth not to name me: and therein most learnedly disputeth with me his jealous opposite, and supposed, and assumed author and divulger, and what not, in sixe whole leaves, as question about the affes shaddow, I should have sayed, whether the ring, or the Index at the center bee the better? that word BETTER crulley wrings him. What, such comparison of ? Such a comparative aspersion of BETTER? Too great and too loose an aspersion: An unsavory report indeed which favours of too high a conceipt of the one, and too great a detractior from the other. Endeavouring what in him lyeth to annihilate and beate downe the waye, which I write upon, and to glory in the raysing up of his supposed owne: thereby not onely possessing men with an untruth, but making ME also ignorant in MY choice, that I should leave unto the world the weakest and imperfect part of the projection of the Logarithms, and leave the best for another to write upon. I never thought when I first writ upon this my invention or my name so to come to the worlds rumour: which may teach Me and others carefulness hereafter (yea and fit it should) how and what We publish to the world: seeing there are carpers and maligners such busy-bodies, who marre what others make: such who have stings like Bees and arrows to shoute; sharp-witted criticks, Diogenes like snarlings, who while they will needes have many callings, neglect their owne. Good Sir be pacified: who troubleth your patience? I, whom you make your adversary (a better friend than you deserve) never I assure you, delivered that compatative attribution; I disclaime it utterly: I never made comparisons with you: you must seeke you some other antagonist. And now what is it had been much better, and more for your honesty, to have held your peace.

But we will goe on in your Pamphlet....


After all this he rambleth back againe by way of introduction of the examination of the graduation of the circles of the ring which may serve as an inducement and furtherance to the learner, to fit and acquaint him. Wat, are we no further yet? we have fairely rowled Sysiphus stone: but to make amends, we have a few scrambling uses in Astronomie, in Dyalling in plaine triangles, from pag 56 to 67. And then the Flag of encomiastial verses of pg 25 is again gloriously displyed in pag 67. Fy upon foulery! Fy unpon vaine-glory! Fy upon such miserable penury of matter!

[ Much, almost 80% in fact, of the Epistle is not currently online here.....]

* References

[1] A. J. Turner
William Oughtred, Richard Delamain and the Horizontal Instrument in the Seventeenth Century, Annali Dell Intituto E Museo Di Storia Della Scienze Firenze (1981) 6, pg 99-125.
[2] Jacqueline Anne Stedall
Ariadne's Thread: The Life and Times of Oughtred's Clavis, Annals of Science (2000) 57, pg 27-60.
[3] P. J. Wallis
William Oughtred's `Circles of Proportion' and `Trigonometries, Trans. Camb. Bibliog Soc. (1968) IV, pg 372-382.
[4] Florin Cajori
William Oughtred, a Great 17th Century Teacher of Mathematics, Chicago (1916)
[5] D. J. Bryden
A Patchery and Confusion of Disjointed Stuffe: Richard Delamain's Grammelogia of 1631/3, Trans. Camb. Bibliog Soc. (1978) VI, pg 158-66.

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