Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks.

To the Reader

HOw undeservedly oft are the single and sinc••re ende∣vours of some men by the malevolent disposition of en∣vious detractors backbited, (w^ch sometimes rebounds backe aversly upon them) not onely by bare assertions, but also by injurious, and contumelious aspertions; In which kind I have not a little lately suffered: for having for a generall end (more then ayming at mine owne particular) published the ma∣king & use of my Horizontall Quadrant, with my new invention of the Projection of Logarythmes Circular by a former booke of the use of it, intituled with this, Grammelogia, or the Mathematicall Ring, since that time I have beene deepely glanced upon, and scandalized both about the former and latter, these detractors taking away from one, and giving to another, famousing some, and infaming others, which did not a little disturbe the quiet and Peace which formerly I injoyed, but did also disorder and slacke my intentions, in the pub∣lishing of the inlarging of the Invention of my Ring, as I promised to doe in the Conclusion of the aforesaid Booke of the use of that Ring, which would serve as a helpe for such as affect Mathematical practises for the working of Trigonometrie unto minuts, and to give accuratly Rootes and Proportionall numbers, unto 5. or 6. places, as is spe∣cified before in the description of the projection, and had ere now beene published with the excellent use thereof, (not out of any Mercinary respect nor interlased with untruths, de∣lusions, and bumbast stuffe, by way of Illustration if not con∣fusion) had not some envious calumniators stole away my inten∣tions, in stealing from me my labours, by detracting from me, and assuming unto themselves the inverting of the Circles of my Ring up∣on a Plate or Plaine, accommodated with an Index to open and shut at the Center, when another imitating thereof, did so fit i••, whose mo∣desty was such that he would not derogate the invention of the projection from me, to himselfe, but ingenuously acknowledged it being so made up, and contrived, to be the same proiection with my Ring, But the ma∣nifestation of it, so first to the world, properly and soly belongeth unto him; which Circles so projected on a Plate with an Index, was not also

unknown unto me, as by good testimony I can produce, before any of thes•• things came to the worlds view, though publikely I writ not first up∣on them: the Learned in those Artes, and those that understand the projection, know it to bee one and the same, and are not deluded b•• supposing a new thing, as many are, for it is but (as a learned ma•• said) as to turne a Garment in and out; A motion must performe th•• operation to give proportionalls with such expedition, and not other∣wise, as in my Epistle to the Reader upon the use of my Ring I hav•• delivered, howsoever the projection on the Plate or Plaine I had pub∣lished ere now, not in so an unprofitable and obscure method as i•• now delivered, had I not beene prevented by some others, whos•• callings might have invited them to spend their houres better, tha•• to snatch with greedinesse that out of anothers hands which wa•• not their owne: for these Circles of my Ring projected on a Plate o•• Plaine, so fitted with an Index at the Center as aforesaid, by cleare testimoney in its particulars I shall prove in his due place to be ano∣thers, and not the supposed Authors, whose conscience may checke•• him and tell him herein, that he never saw it, as he now callengeth 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. & not the Sommer before, as is falsely alledged by some one who hath made himselfe a spokeseman to another in some things by equivocations, & in other things by confessed untruths, whose ambition to be some body, hath incited him forward to deliver some supposed new stuffe, or scram∣bling peeces, if not confused fragments of his owne, or some others, to a publike view, in obscure and various phrases, a thing supposed to bee forged by sundry heads, rather than by one alone, seeing there is such roving from the Text; amongst whom to blow some smoake thereto, there was some grosse one, seeing the matter is so com∣mon; for to a finer element perhaps his capacitie could not assent, or ascend. A blinde Guide and a Parrats speech are not much different, the one walkes hee knowes not whether, and the other speakes hee knowes not what; and such are all precepts in Artes, which leade and make men speake without Demonstration; which doth not onely protract the studious, and frustrate the affectionate, but makes an ingenious spirit (who ever is more Rationall th••n practi∣call)

to contemne such Circumlocutions, and laugh in private, if not in publike, at the learned stile of some Authors, who ma∣king themselves by their obscure kind of writing seemingly fa∣mous, sticke not to calumniate others to make them infamous. Those that are but initiated unto knowledge, for their faylings and defects of order, are worthy of some blame, but for others who would so are above all, and not onely picke holes in the coates of the living, but also vilifie the dead, tis a shame; making too great hast at the begin∣ning to glory in that, which no doubt will prove shame in the ending. Its a common thing, that, one man having laboured, planted and sow∣ed, with great paines, another reaps his Harvest with no indu∣stry; yet in this there was some honesty shewen, not to take the Crop but the Gleanings, houlding it easier to follow a beaten path then hazard a discovery, but the way was not made plaine, and the vaile remooved to helpe his sight: God that gave me the former in∣vention without the advise of any, hath also reserved for me the ma∣nifestation of the latter, without the helpe of any, which I formerly mentioned in the Conclusion of the use of my Ring, to declare as is aforesaid, and I hope no envious, and insinuating detractors, will hereafter assume this also to himselfe as his owne, and say I had it from him: I have hitherto borne the injury of the infamy with great greefe of heart, (and God that is the discerner of the spirits knowes mine integritie and innocencie herein.) The window hath beene as yet close, and darkenesse possesseth the place, I will now with∣draw the Curtaine that the Sunshining light may appeare, to expell those 〈◊〉〈◊〉 mists that have beene scattered, and by a true and sin∣cere medium remove that which was suggested from a false. Many have spurred me hereto, who suffer also with me, who wonder at my slacknesse, and long patience, others contrarily have as much gloried, I hope in the end that truth will burst forth, that God may have the glory, man the shame: and I doubt not but such as men sow unto others, such shall they reape unto themselves againe, and with what measure they meate, the same shall be meated unto them, I will there∣fore in the first place answer for my Quadrant, then afterwards for my Ring, and lastly for my selfe, and others.

The Answer upon my Quadrant.

I Have not onely suffered by way of divulging of the said project in of Logarythmes Circular, 1630. but also by the late publishing of my

Horizontall Quadrant, 1631. by a scandalous aspertion couched un∣der the relation of the Author, in the words of his Translator (but ra∣ther his Transcriber, if not in the most of that Booke as is suspected, his Compactor) which are as followeth, Which whilst I went about to doe, another to whom the Author in a loving confidence discovered his intent, went about to preoccupate, and prevent, if not Circumvent. Now because his words are cautelous and subterfugious, we must a little examine them; If they be true, then that which I have produced and delivered as mine owne cannot stand, well then, this preoccupa∣ting, preventing, and circumventing, and discovering in loving confi∣dence his intention to me, must be eyther about that of my Horizontall Quadrant in the making and use thereof, or touching that of my Ring; the former of which I lately produced before the supposed Author did write so publikely upon the same Projection, as he hath now done in his booke named the Circles of projection, which tract is so farre from being answerable throughout, to that which it promiseth in the be∣ginning, that it seemes rather to pussle the studeous then in any wise to further them) of which I say no more, but advise the studeous Reader onely so farre to trust as is agreeable to the Text, and a true Doctri∣nall Method, which being therein omitted, it doth not onely sur∣charge the memory of the learner, but doth much more frustrate and delude the Ingenious, by a labyrinth of tedious Rules, and ambiguous precepts, when few might serve demonstratively, making them speake Parret like, which would be as little vendible as it is abstruse, were it not thrust on men; I would be loath to put an untruth upon it or him, or any other; it were unmannerly, howsoever I will prove that concer∣ning neyther of these Instruments, to wit, that of my Quadrant, or that of my Ring, the said Author did discover his intent unto me, either in whole, or in part, in a loving confidence or otherwise. And for the ma∣king of my Quadrant, I could not circumvent him whom I knew not, for I drew the Proiection long before ever I heard of the Authors name, (as in its due place I can prove) but this aspertion no doubt was from an inveterate hate of some, who endeavoureth not onely to annihilate my interest in the Invention of that Ring, and Quadrant, but also to bring me in a disrepute, and to leave a blot of infamy up∣on my Name unto posteritie; But I doubt not in a publike Audience I shall cleare my selfe of it, and the disgrace that was intended to be cast upon me in the beginning, may light upon the contrivers thereof in the ending, and as there was a pit digged for another, perhaps the diggers may fall therein themse••ves, let them laugh on, as they have

begun while I hasten the Issue of it. The extendure of Gods hand in his donations is manifould, and where his spirit pleaseth to breath, there a dore is opened; now whether the gift to the creatures bee di∣vine or humane, we should blesse God as the first and principall Au∣thor, and giver of all, no wise d••spise man, as the second agent, or receiver: Its very phantasticall in some therefore, who thinke such and such things are not worthy the generall vote and allowance, if they proceede not from such, and such an one, envying they should be produced by any other; which if they be so divulged, as much as lyes in them, they will hasten to possesse the world with a contrary opinion, thereby wronging God in his dispensation, and man in his reputation, but such men wedded to a private and deluding fancie, chuse rather to abandon the Lore of sound reason, then to be divorced from their prejudicate affections. I will in the first place therefore make way for my Quadrant, that I did not circumuent the said Author for the attay∣ning thereof, (as before:) But whereas hee taxed me with going a∣bout to preoccupate, or prevent him, it seemes somewhat soaring like, to abridge any mans free affections, in constraining them to waite on any others concealed intentions, which they never knew (in which every man hath a freedome to himselfe) It is (no doubt) somewhat too malepart, & too rigid, to ty the liberties of others in their actions, who desiring the good of others in not concealing things, if they shall by their industrious labours, shew the excellent use of such or such a thing, to a publicke view, for a generall benefit, are for it envied and calumniated: which as it smells of too great a detracting from others; so it hath its s••ource from the philautie, and too high conceat which some entertaine of their owne worthines: the said Author having had the Projection 30, yeares before by him (as he giveth out) wherefore did hee not publish it then, or give way to it as hee hath now done, how could it bee so long concealed, and others never so much as heare of such a thing, was it that hee would not have the same communicated to others, or that hee would not be knowne in his name by a profitable action, or that some others might challenge an Interest in it besides himselfe, or that the uses of the Projection lay hidden, and obscure; or that they were not at that time so plentiful∣ly made manifest, untill M^r. Gunters ingenuitie opened the mystery of it, and applyed that projection so methodically, and copi∣ously, to Horolographie in his Booke of Dyalling in the use of the Sector, and accommodated that Projection long before in a Dy∣all for an honorable personage, as the first that ever was made in

that kind, and the same forme with these that are now made, and therefore to give every man his due, and not to iniure the dead, it is properly ••••en M^r. Gunters Dyall, for that composure, and not ano∣thers, (notwithstanding the inverting of things and detractions from him) But it should seeme he would publish it then, eyther when the way was made faire for him, or when he might catch at some one, as lately hee hath done in an unfit and uncharitable way: And as M^r. Gunter was copious in applying that projection to that particular of Dy••lling, so might he no doubt have beene in the Astronomicall uses of it also, though he delivered so few as but six observations one∣ly to a publicke view upon that Projection, which before the publish∣ing and also after, both in the uses and the Projection it selfe I often intentively looked upon, and extracted from them many usefull per∣formances more, even in M^r. Gunters time; and since his death have published my Horizontall Quadrant, extracted from that dyagramme: in which I have abundantly supplyed the emptinesse and obscurity of that Projection, fitting it for a Pocket Instrument, or according to any magnitude, as a helpe and benefit to those that are studious of Mathe∣maticall practices; which labours, of M^r. Gunters, and my owne, if they were not to unvaile the subiect and to make way for ano∣ther to helpe his sight, which writes afterwards, I referre it to an eye not partiall, to be Iudge, it being rare and wonderfull for one man to see all at once, and there are farre more excellent uses yet upon that Projection, which may be also knowne if some one will open the vaile a little more; if there be any that knowes further uses upon it, let them discover them, for the present use for others, and let them not, if others, more respective of the common benefit of such who delight in those things, publish them, by scandalous detraction deliver that they are prevented, if not circumvented. Which Quadrant in the originall as before specified I did extract and compose from the fun∣damentall dyagramme of M^r. Gunters Booke of the sector, page 66. as I have specified in the Epistle to the Reader, in the booke of the use of Quadrant, howsoever the supposed Author (having the free liberty granted to see my Epistle before it was printed, and alter whatever he thought fitting, (I being unwilling to oppose his desires) did dash out whatsoever I had there to my knowledge justly attributed to M^r. Gunter, because he said it did belong to him primarily, yet I must say that I was especially behoulden to the labours of Mr. Gunter who is now at quiet in his Grave, and therefore not to bee wronged. And should I search the originall from the first, neyther M^r. Gunter nor

especially hee may challenge so much unto himselfe, since the maine draught out of which theirs is extracted, was extant before they were borne, namely in that auntient Geographer Munster in his Dyalling, where it stands obuious, as also in the famous 〈◊〉〈◊〉 Oronti∣us, as I mentioned in my former Epistle, but I suffered him to put it out, being (it should seeme) unwilling to have his owne dis∣mantled; As also at that time, I shewed him in our English Blagrave the like Scheme, in both their workes, long ere hee or M^r. Gunter committed any such thing to a publike view; howsoever, I had not the least touch of furtherance from him, or from any man breathing, either by transcripts, or verball direction, (but onely what I have formerly acknowledged out of M^r. Gunters Booke of the Sector, page 64. 65. 66. though in other triviall matters I doe and shall ac∣knowledge freely) as hath bin falsely alledged by some loose tongued Instrument, that I had the making of that Quadrant by a sight of the Authors letter, whose honest relation may bee suspected, seeing hee makes so little conscience, to detract from anothers reputation whose detraction is grounded upon as great a stabilitie, as vncertainty, whose Basis is onely a bare supposition, conjoyned with an indisposed evill eye, as full of envie, as emptinesse of charitie, and this upon examination he is able to make good, and no more, which a gene∣rous or tender breast scornes to harbour, much lesse so partially to divulge: But it will no doubt make such Tale tellers so much the more odious to the sweet disposition of such noble spirits, when they shall be possessed with the manifestation of the contrary, which will no doubt shake the foundation of that Projection, by letting the world know it from the originall, that he which hath given way and gloried at the countenancing of that aspertion, and as the supposed Author of that Projection, perhaps may be challenged by others, as I mentioned before, howsoever therein very ingenious, by so assuming the Projection to himselfe (as in his owne words) but not ingenu∣ously enough acknowledging from whence he had it: for my parti∣cular, I take God to witnesse I have without any equivocation or men∣tall reservation, declared in every particular the very naked truth, in the Epistle to the Reader in that Booke of the aforesaid Quadrant, from whence I had that Quadrant, and how I produced it, and it might have bin easily so composed, and published, as I have done it, by another, (without the helpe of the sight of a Letter,) that is but indif∣ferently versed in delineations, seeing the Projection in my Author M^r. Gunters Booke, page 65. 66. as aforesaid lies so plaine & conspicu∣ous

(notwithstanding) in which letter to my remembrance I saw bu•• some ordinary uses, and a checke cast upon Mr. Gunter, but no di∣rections in it for the making of the Instrument; which uses I slighte•• as meane, and triviall, and other things I saw not: and this I speake not to shuffle things off, but out of a true sincerity, but perhaps since the sight thereof there may bee inserted somewhat else, to make my opposites assertion good, howsoever I needed not such a helpe by the sight of a Letter, seeing Ioyners, Carpenters, and other Mechanicks about this Towne & else where, yea, Schoole Boyes in imi∣tating the Projection aforesaid in Mr. Gunters Booke of the Sector, and following the directions therein in the 66. page of the use of the Sector aforesaid, have drawne the Projection fully and compleate by the Booke alone; and others having onely had but a simple view of my Quadrant, many yeares before I published the use thereof, have from the aforesaid Booke of the Sector page 66. made the like, having not had the least assistance from any, but the direction of the Booke onely, as upon oath they have beene examined, and doe acknow∣ledge, and will testifie it when occasion requires: beside I know sun∣dry Gent. and others in this kingdome, that are yet living, that have drawne the same Projection by the Booke, alone immediatly after the publishing of it 1624. as at this instant they wil be ready to confirme, and my selfe in Mr. Gunters time 1622. (besides many others) have drawne the same Projection, for our particular uses, and are yet to be shewen, w^ch was long before either I, or they ever heard of a new Au∣thor, of that Projection besides Mr. Gunter onely: And to make my assertion yet more absolute, I did not onely draw the Projection in Mr. Gunters time, but before his death did also shew the making of it to others; therefore (as before) I could not circumvent the said supposed Author, to have any assistance from him in making of my Quadrant, either by a verball declaration, or by the sight of a letter: or otherwise, which in its due place God permitting life, and health, shall be confirmed more at large in every title.

Ʋpon my Ring.

IF I circumvented not the said Author, nor that in a loving manner he opened his intent unto me by assisting of me in the making of my Horizontall Quadrant; Then to make his assertion good, it must ne∣cessarily be in the other Instrument, which I produced and published, to wit, that of my Mathematicall Ring (that is, how I might compose

or make the same) but concerning this latter Projection of the Ring or any thing to that effect, in a loving manner then, or at any time, or otherwise that he discovered his intent unto me, it can hardly bee col∣lected (certaine circumstances seriously weighed, and considered) The whole ground of which being from as weake a principle, to open the way unto me for the making of my Ring, as the sight of a Letter to shew me the making of my Quadrant; for about Alhalontide 1630. (as our Authors reporteth) was the time he was circumvented, 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 Theoricall 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 you say to an Invention that I have, which in a lesse extent of the Compasses 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 Archingwise, 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 immediatly I answered, I had the like my selfe, and so we discoursed not a word more touch∣ing that subject: all which sundry times ingenuously according to the very truth the said supposed Author hath acknowledged before divers persons, who doe and will testifie the same: Then after my com∣ming home I sent him a sight of my Proiection drawne in Pastboard: Now admit I had not the Invention of my Ring before I discoursed with the supposed Author thereof, it was not so facil for mee or any other (to an eye not partiall) to raise and compose so complete, and absolute an Instrument from so small a principle, or glimpse of light, but was knowne unto me, (as I have produced it) long before that time, and being now published as it is, the composure of it seemes most facil, (as all inventions doe, once knowne) that I have much wondred with my selfe, that Mr. Gunter or some other produced it not so to the world as now it is, seeing it was so easie, and caries with it such an excellency, above that which is in the Lineary forme, for in a

Circle it is naturall, & perfect, in a line defective & imperfect: But he, or they, perhaps saw it not, though their sight I confesse (no wise to dispa∣rage their worth) might penitrate further in other things; but gifts may not be attributed to naturall ingenuitie, but to God the giver of them, who disposeth where his goodnesse pleaseth. And who knowes to the contrary but many private men in this kingdome, or elsewhere, might have this Invention long before my selfe, or he that now chal∣lengeth it, seeing the gifts of God in his donations, to severall per∣sons, are oftentimes in one and the same thing, and if God bestowes his Talents on us, we ought to communicate them to others, which not to doe, is to hide them in a Napkin, and who so concealeth them are not free from some reprehension: our Author may answer for himselfe, who reporteth now that hee had the Invention for more then twelve yeares past, but put it not to use, which in a sense was in∣jurious both to God and man, if hee saw the copious, and compendi∣ous use of it then, as now hee doth; to mee and to many others it hath seemed strange, that he should hide such an excellencie, so long in obscuritie; But it is supposed, that hee had then but a glimpse of its performance (untill I writ upon it, by opening the Cabenet, and shewing its treasure to the world) and so regarded it not then as a Iewell, but since. But 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 car∣ry with it so delightfull & pleasant a taste, knowing that as the Loga∣rythmes (as a Iewell) that did excell all Inventions that ever went be∣fore it, so this kind of Projection, as the daughter springing from so noble a Mother, was the rarest Instrumentall Gemme, that ever Art in all preceeding Ages did afford: which in its production heretofore lay obscure, but now being published how cleare doth it seeme to bee in its composure, and if it were so easie at the first to produce (as some account it) why did not some one all this while since the pub∣lishing of it (being now more then two yeares past) inlarge that Invention, or deliver others in that nature of their owne, being easier to adde unto an Invention, now being divulged, then to deliver its ori∣ginall? But hitherto it hath seemed difficult, many having attemp∣ted and endeavoured, but have fayled in their ends, howsoever I know not what some private mens industry hath produced by way of aug∣mentation to that invention of my Ring, or the Logarythmes Pro∣jected Circular: At the Conclusion of my Booke I gave a sufficient and perspicuous direction, and invitation thereto, that so by the for∣mer

labours, and the other there intended, the way might be made faire, and the Ice broken that they might wade the easier: and so produce at last something for a publike use by Augmentation to this Invention, at which I shall much rejoyce, rather then any wise en∣vie: But I have as yet all this while demurred in my pretended pur∣pose, and being often times by sundry men importuned, according as I have promised, to inlarge that Invention, seeing none as yet hath done any thing therein, I have now at last therefore for their benefit, & others, to a publike view delivered what is mine owne to that end: and that by sundry wayes, and how easy doth the inlargement of that Invention now also seeme to be, being now produced. With the like facility did I compose long agoe my Helicall line of Roots which affords five places accuratly, by an inspection of the eye onely, which being not as yet divulged to a common view may to some seeme difficult for its composure: The secrets and intentions of the mind, how closely are they lodged in the breast, and who can search the heart, that from a word barely delivered the whole may be con∣ceived; therefore, it was not easy for me to know the Authors intent in his project by a bare word, had not God long before opened the way unto me, how this projection, or Invention might be composed; what meanes then these words; that in a loving confidence the Au∣thor discovered his intent unto me? is it not to give a faire flourish up∣on a untrue subject, to delude, if not to possesse Men with a falsitie, to detract from another his good name; besides the barrennesse of the word we will a little consider the straightnesse of time, in the original, for the producing of the Booke, and Invention, as is challenged by the Author to be about Alholantide, 1630. & that then his loving intent to me he opened, and so was circumvented; I will put my cause into the hands of any indisterent Iudge to censure of it, that having had no other direction, or light from the said Author then formerly is speci∣fied, (for more hee dares not avouch with a cleare conscience) how could I from it so easily forme my Projection: Mr. Gunters Ruler, as some thinke, was a furtherence to me, but it was rather a hinderance, for his line of Numbers was as impertinent for me to follow, as such a double composure in the projection was superfluous, therefore such a bare verball (as before) or Instrumentall dictate was not used, or could be sufficient to compose so high a worke: but it was from a long intentive precogitation of many yeares with my selfe, how the Logarythmes in the Tables might be so compacted, that all Num∣bers in these Tables should bee proportionall one unto another, and according to a divers, and variable proportion assigned; which to

effect I found at the first very difficult, and could not conceive how otherwise it might bee done, but either by fitting of Tables to all proportions, which so to doe would not onely be too great for opera∣tion, but also breede confusion; or it must be from some graduated in∣sertion of these numbers Instrumentally, so that by motion numbers might be mooved one unto another; and for expedition of which I found no figure more apt then a Circle, and on that my Meditati∣ons fixed, and there I rested, and so (as my few houres could permit me) I made severall projections, that the senses might see the effect of it in a perfect beauty, which the intellect saw before but in obscurity: about which how often was I afterwards interrupted in my desires to looke further into the mysteries that lay open in that new Invention, having not scarce an houre in a day, and sometimes not two houres in a weeke of serious privacie (by reason of my calling) to sport my selfe in operation thereon (for it seemed so to me then as a recreation, as all new Inventions at the first doe to any that invents them) It was many moneths therefore before I made tryall of it, in the generall uses that it might bee put unto, in matters of Arithmeticke, for common o∣perations, and to the measuring of Plaines and Solids, but especial∣ly how it might be applyed to the doctrine of Plaine, and Sphericall Triangles, in Astronomicall Calculations, Nauticall practices, and Ho∣rologicall conclusions, all which in every particular I practised on it at such times as convenience would permit mee; And having for ma∣ny moneths thus satiated my selfe, and fed my fancie upon its Theo∣ricall contemplation, and its Practicall operation, though often∣times I found many Rubs and impediments in the Practise, in ap∣plying this instrumentall Invention at the first so generally and co∣piously as I did; And it cannot bee denyed of any, that the wayes of new Inventions lies not so obvious, or so easie, to bee discovered with such celerity, as a long premeditation might produce; I was desirous therefore to make the world participate thereof also; Therefore at severall times, having but a little time at anytime, by little and little did I compose and produce a method pleasing to mine owne fancie; doing one weeke such a peece, another weeke another, and so going on untill I had runne through the whole parts of Arithmeticke, in the Goalden Rule direct, and indirect, in division and in multiplication, applying all these to manifold uses in combination of Numbers, to common affayres, in fortification, to mensuration, and fractionall ope∣ration, then I applyed the Instrument to the finding of numbers in continuall proportion, in finding of meane Proportionalls, and the

extracting of Rootes, I laboured further to make the Instrument more complete that it might worke all usefull proportions touching Interest, or valuation of Leases: and last of all I applyed the Instru∣ment in the Circles of Sines, and Tangents, to some uses of Astrono∣my, in the finding of the Declination of the Sunne, and its ascention through all Latitudes; These things I drew up in a Booke at severall times aforesaid, as a part of the practises that I made upon it: And it was a task sufficient for a man that had cōmand of his houres, by allot∣ting them soly to that end, in as long a time as was betweene Alhalan∣tide & Christmas, in the opening of the way of that new Invention, & by applying of it so, (to use for a publike view) rather then by one that was commanded by them, especially at that time and all the Tearmes before: for how few are the houres that a man of any imployment gets to set upon a new method for a new Invention at such a time, I leave to any indifferent eye to judge; and it would have beene so much the more harsher, and difficult to have it so suddainly produced, had it beene then at Alhalantide but onely conceated, and agitated on, but it was premeditated on long before, as is aforesaid, and in∣tended for a new yeares gift for the King, which accordingly I gave him on new yeares day, though a fortnight, or three weekes before Christmas, his Ma^tie received from mee a Scheme of the Projection in Pastboard, with a manuscript of the Booke which is now published, agreeable to it in every tittle, (the Epistle excep∣ted) a Copy of which was then at the Presse: and was Printed foure dayes before new yeares day, 1630. Now that all these Practi∣ces, and many Transcripts that were drawne, with the doubts and hinderances that did arise in the fitting of these things, could bee made, so ordered, and produced in so little a time, as the scattering of a few houres as betweene Alhalantide and a moneth before Christmas, I leave to the Iudicious Reader to censure: I broke the Ice, and made the way facil for another that came behinde mee; yet if hee tooke a yeare and more to meditate and write upon my endeavours after my publication (thereby not to sig∣nifie that hee wanted time, but takes liberty enough if not too much, to the losse of time) as long a time, halfe of it, or a quarter of that time, may by a Charitable boone be granted mee. Which consideration of time, will easily also cleare mee from the imputation of the Authors loose assertion, that hee was Circumvented by mee, nor that in a loving confidence he opened his intent unto me. Yet in the last place I will hasten

to vindicate his untrue declaration before the Courts of justice; if re∣stitution be not speedily made, where true witnesses shall bee pro∣duced, and that which is now but in agitation I will bring to action, and prove that before Alholantide or very neare that time, my Invention was produced to a publike view; therefore it was very injurious in the Author to possesse the world with so an untrue asperti∣on, upon a bare supposition, in that I should have the Invention frō him, and that in a louing manner he opened his intent unto mee, and was Circumvented, as before, whose assuming disposition hath not onely bin busie to take from me my good name, & labours by this his loose aspertion, but hath also endevoured (by too rigid and generall a censure sparing none in some sence) to bring all in a common disre∣pute in their Callings, therefore,

In the behalfe of vulgar Teachers & others.

BY way of advertisement to you, and my selfe, & to whom it may concerne, vulgar Teachers of Mathematicks about this Towne and kingdome, as you are styled, which had bin faire (if no worse) seeing you are not professors, or publike Readers, but common Teachers, it be∣hooves you, and me, that we indeavour to avoyd the disreputation of a scandalous attribute in our profession of so noble a science, that our doctrines be not onely Practicall, but also Rationall, and Theoricall, that we may not bee ranked with Iuglers, and teachers of Tricks, as we are lately glanced upon publikely, but a charitable breast how∣soever (I perswade my selfe) hath a better opinion of us, that accor∣ding to the Talent that God hath given unto us, out of the Riches of his bountie, no doubt in our callings we use them rightly, and doe not (by deceiving) derogate from the end, which is, to glorifie God in these gifts in a true and sincere use of them; which otherwise would not onely be a blemish and staine to our name but also a disho∣nour to our noble profession, the large testimony of which (no doubt) every one of you in his particular can produce, (notwithstanding such an uncharitable censure) can amply testifie in your callings your suffi∣ciency, and therein remoove all scruples to confirme your integritie; and as the imputation hath lighted heavy upon us, so hath it not rested there, but rebounded likewise (if the words be truly scanned) unto such Nobility, and Gentrie, to which we have beene servants to or in present are, in so honourable and laudable a service, which asper∣tions

are as highly backed with arguments, as he was to forwords that divulged them: whose judicious and censorious eyes, hath beene two much busied to shew the wayes of the Iugling of others, and prove delinquents in the same things themselves, whose words are these, That it is a preposterous course of vulgar Teachers, to begin to teach with Instrument, which was not onely to despight Art, to betray willing and industrious wits to ignorance, and idlenesse, but was also losse of precious time, making their schollers doers of tricks, as it were Iuglers. Which words are neither cautelous, nor subterfugious, but are as downe right in their plainnesse, as they are touching, and pernitious, by two much derogating from many, and glancing upon many noble perso∣nages, with too grosse, if not too base an attribute, in tearming them doers of tricks, as it were to Iuggle: because they perhaps make use of a necessitie in the furnishing of themselves with such knowledge by Practicall Instrumentall operation, when their more weighty nego∣tiations will not permit them for Theoricall figurative demonstration; those that are guilty of the aspertion, and are touched therewith may answer for themselves, and studie to be more Theoricall, then Practi∣call: for the Theory, is as the Mother that produceth the daughter, the very sinewes and life of Practise, the excellencie and highest degree of true Mathematicall knowledge: but for those that would make but a step as it were into that kind of Learning, whose onely desire is ex∣pedition, & facilitie, both which by the generall consent of all are best effected with Instrument, rather then with tedious regular demonstra∣tions, it was ill to checke them so grosly, not onely in what they have Practised, but abridging them also of their liberties with what they may Practise, w^ch aspertion may not easily be slighted off by any glosse or Apologie, without an Ingenuous confession, or some mentall re∣servation: To which vilification, howsoever, in the behalfe of my selfe, & others, I answer; That Instrumentall operation is not only the Compendiating, and facilitating of Art, but even the glory of it, whole demonstration both of the making, and operation is soly in the science, and to an Artist or disputant proper to be knowne, and so to all, who would truly know the cause of the Mathematicall operations in their originall; But, for none to know the use of a Mathematicall Instrumen, except he knowes the cause of its operation, is somewhat too strict, which would keepe many from affecting the Art, which of themselves are ready enough every where, to conceive more harsh∣ly of the difficultie, and impossibilitie of attayning any skill therein, then it deserves, because they see nothing but obscure propositions, and

perplex and intricate demonstrations before their eyes, whose unsa∣voury tartnes, to an unexperienced palate like bitter pills is sweetned over, and made pleasant with an Instrumentall compendious facilitie, and made to goe downe the more readily, and yet to retaine the same vertue, and working; And me thinkes in this queasy age, all helpes may bee used to procure a stomacke, all bates and invitations to the declining studie of so noble a Science, rather then by rigid Method and generall Lawes to scarre men away. All are not of like disposi∣tion, neither all (as was sayd before) propose the same end, some resolve to wade, others to put a finger in onely, or wet a hand: now thus to tye them to an obscure and Theoricall forme of teaching, is to crop their hope, even in the very bud, and tends to the frustrating of the profitable uses, which they now know, and put to service, and to the hindering of them in their further search, in the Theoricall part, which otherwise they would apply themselves unto: being catched now by the sweet of this Instrumentall bate; which debarring would not onely injure the studeous but also cause the Mechanicke workemen of these Instruments, to goe with thinner clothes, and leaner cheekes.

Neither doth the use of Instrument to a man ignorant of the cause of its operation, any wayes oppose or dispite Art, seeing that the end of producing and inventing of Instruments, is their Practicall use; Be∣sides, its impossible to shew the use of an Instrument but in teaching there must needes be laid downe some grounds or prolegomena••, as, what is meant by such, or such names; what are such, and such tearmes, and therefore the beginning of a mans knowledge even in the use of an Instrument, is first founded on doctrinal preceps, and these precepts may be conceived all along in its use: and are so farre from being exclu∣ded, that they doe necessarily concomitate & are contained therein: the practicke being better understood by the doctrinall part, and this later explained by the Instrumentall, making precepts obvious unto sense, and the Theory going along with the Instrument, better in for∣ming and inlightning the understanding, &c. vis vnita fortior, so as if that in Phylosophy bee true, Nihil est intellectu quod non prius fuit in sensu, and things the more they be objected to the sense, are more fully represented to the understanding, then it must needes follow that a doctrinall proposition laid first open to the eye, and sense, and well perceived enters more easily the dore being opened, then if the intellect by the strength of its active sense, should eliciate, or screw out the meaning by a long exeogitated operations; Its not therefore requisite unto all Capacities, to have Instrument severed from science,

in many things: is though the use of one could be without the other lesse or more.

Neither doth the practise on them betray willing & industrious wits to ignorance and idlenesse, that assertion therefore is very ridiculous, for he that is industrious, and willing to spend his houres, for the at∣tayning of any science, according to doctrinall method, busieth him∣selfe not with Instruments, but applies himselfe to such Authors, or such compendious abstracts, as are taken from them, which doe not onely open the essentiall parts of the subject in the Theory, but also layes downe such documents and principles which may (in a higher nature and way) induce him to practise them, then possibly by Instrumentall operation for exactnesse can bee attayned unto; for such is the excellency of Art by Theoricall doctrine, that all things tending to practise may bee done by the science onely, without the helpe of Instrumentall operation, certaine propositions being granted, which originally and principally are proper unto In∣strumentall observation, being the Basis or foundation of the whole; Therefore science, as before, hath a principall depen∣dance on Instrument, but is it in their observation, rather then in their operation, and the Inventions that are daily produced in that kind, are onely to compendiate and facilitate practicall things, which the learned in those Arts (having the science) scarce use at all, if at any time they use them, it is in small and triviall things to satisfie the sense, and not the intellectuall part; which The∣oricall way doth not onely augment the desires of the Teachers in the accommodating things, and remove of difficulties, by making them conspicuous to the learner; But also by so proceeding causeth the Teacher to adde many ends both pleasant and usefull, by the way, for the Practicers incouragement. To begin with Instrument is unprofitable for the Teacher, though advantagious to the learner, (if his ends be but to know some uses upon an Instrument, as it is with many) for it is easier for a man to learne more usefull Practise upon some Instrument in one houres Instruction, then to know the cause in 20. houres of some Instrumentall operation: and yet there are many Instruments almost as facil in their demonstration as in their opera∣tion; Which kind of beginning to teach, or usuall proceeding there∣in is not as vulgar Teachers use, but is as indirect in the Method as false in the assertion (if they may possibly avoid it) for that were to teach against their owne profit, and the dignitie of learning; Yet that any Teachers of Mathematicks should be so nice as to deny the use of Instrument, to such Gent. or others, who perhaps desire the Theory to

contemplate on hereafter: but the practicke for the••r private ends, for the present, were not onely to frustrate their desires, but injure their occasions; who might rather laugh at the teachers pride, then contemne his Art.

Lastly, that the practise on a Mathematicall Instrument should be losse of precious time, (to any one that knowes how to use them in their practise though not in the Theory) is not onely ridiculous but also untrue and absurd, for to what end did the manifold and laborious calculated Tables of the learned in their Subtenses, Sines, Tangents, and Secants serve, were they not to avoyd the great toyle of Alge∣breticall worke in Radicall extractions, which otherwise in Trigo∣nometrie (the practicall end of Theory) must necessarily be used, but they were Invented & produced to avoid the losse of time, to the more learned (who know the causes of their operation in the Genises) and as a facilitie to open the way unto the unlearned, though thousands working Trigonometrie by them knowes not scarce the cause of such operation, which in a manner is as meere Mechanicall, as if it were Instrumentall (though more accurate in its performance) But to come nearer unto these present times, how hath the Invention of Logaryth∣mes, taken away the labour and losse of time, that was used in former calculation, for that which cannot be done by the common Tables of Sines, Tangents & Secants in 20. hours, is now done by the helpe of the Logarythmall Tables in one houre, & there are thousands in the world that also worke by them for their private use, by reason of their great quicknesse, in hard and laborious matters, who know not the cause also why these Numbers should so expedite such a difficultie, with so great a facilitie, & celeritie: yet they use them though the performance be also mechanicall, that they may not likewise lose time: Now In∣struments though they bee extracted from Tables, yet according to their capacitie in practise they exceede these Tables, in all common serviceable uses, that I dare maintain that what the Logarythmes by the Tables, for giving of proportionalls, either in Numbers, Sines, or Tan∣gents, will doe in 20. houres, my Ring or the Circles so projected from it, will doe in one houre. Which Instrumentall Invention, I have also produced to helpe the studeous to avoyd the losse of pretious time, which to omit and keepe them from it (in some occasions) were also misspending of time: Therefore in respect of private mens ends, to be∣gin with Instrument is not to Iuggle, nor to doe tricks, neyther to oppose or dispite Art, or to betray willing and industrious wits to ignorance and idlenesse, nor losse of pretious time.

BVt I would be loath while I seeme to vindicate the losse of time, cause others to lose it, by my too prolix defending of it; the vindicating of the truth indeed was my cheefest ayme, which I hope will be my spokeseman & defend it selfe, meane while not to hould my Reader too long, I wish with all my heart that such occasions were taken away, then might he spend so much time in reading better things, and I in pro∣pagating, not in defending my right; detraction, Calumnie and defamation come close to a man, and therefore my Reader may excuse me if I be so tender of it, since it may be his owne case (how innocent soever) perhaps the very next of all. There is (and I am not the first nor it m••y be the last truly sensible of it) a generation in the world true sala∣manders whose delight is to live in the fire of con∣tention, endeavouring to loade others with Calumnie and detraction, and to raise unto themselves a proud Babell out of others ruins; a vice odious both to God and man, there∣by forgetting the rules of equitie which ought to be obser∣ved, so to deale with others as one would be dealt withall, thereby blemishing the good name of another, who are not ignorant (or at least should not be) how pretious the name of a man is: is it not above Riches, above pleasure, aboue Gold it selfe? Calumnie indeede is the fire this Gold is cast into sometimes, onely heeres the difference, this puri∣fies the one, and that strives to pollute the other; nay, what speake I of those grosse things, is it not dearer then life it selfe, being the very life of the life here, and eternitie of a

mans life hereafter? If he that steales but a small matter from his neighbour is obnoxious to punishment, sometimes unto death, is not he much more worthy of the same, and greater punishment that layeth violent hands upon that which is above life it selfe? If reports were true, private admonitions may win men, if false why are they then divul∣ged? and grant the reports be true yet, is every report to be spoken? may not a man breake the ninth Commandement that divulges these reports though true, which tend to the infamy of another? and how much more is he guilty of this breach, that is not onely a blazer of such reports, but a compiler? Is not Satan stiled from this word of Calumnia∣tion? doth he not Calumniate God to man, and man to God, and one man to another? and shall any man tread in his steps? is it not the precept of the Psalmist, or ra∣ther of God by the Psalmist that every one should set a watch before the dore of his lips, and keepe in that unruly member, that is, a world of wickednesse? what is the stroake of the hand to the stroake of the tongue? the one wounds the body, the other wounds the soule; the one may be easily cured, the other hardly; the one strikes them that are present, the other those that are absent; the one striks but one at a time, the other strikes many at the same time, at least three at once, with the selfe same blow; viz, him whom he doth traduce, him who heares him, and himselfe, and many times the last most of all; and ought wee not to take great care how we order this unruly tongues what Gall, what Poyson, what Wormewood, what Wildbeast is to bee compared with the tongue, which lancinates, teares, and wounds the soule, by the virulencie of its words, and reports: Looke into the heavie judgement of God against such, 1 Cor. 6. that

revilers shall not escape punishment. Such was the account of the preciousnesse of a mans goodname in ancient times, that there hath beene Lawes utterly to banish such from the society of men, who by their Calumny and detraction sepa∣rate all good societie amongst men, making foes of friends, and breaking the sweet bands of peace and charity, cau∣sing nothing but brawles and dissention, and as neare as they can dissolving this goodly frame into the ancient Chaos of confusion. Againe, some of the Antients herein are so vio∣lent that they call it Grave malum, Turbulentum Daemona Pestem pestiferam, and professe plainly, Qui detractioni stu∣dent diabolo serviunt, so that as Coales are to burning Coales (saith the wiseman) and wood to fire, so is a con∣tentious man to kindle strife The words of a Tale bearer are wounds, saith he, they desturbe a mans Peace and goe downe into the inwards of the belly, and a whisperer sepa∣rateth deare friends. I wonder what neede these Lamiae gad so much abroad having worke enough at home, Te prius corrigas quam alterum corrigas, If such would turne their eyes inward they should finde matter enough to mende in themselves first, and afterwards they might reprehend o∣thers, sed ad delicta nostra Talpae, ad aliena serpentes su∣mus. It was once Constantines speech. That hee would throw his mantle over his brothers nakednesse rather then lay it open; What shall wee thinke meanewhile of those who would faine discover nakednesse, where they can finde none? what should bee the plot, except it bee to feede the bosome Wolfe of envie and malice which at last will gnaw out their owne bowels? doe they envie be∣cause God gives to one man one gift, to another man another? would they have all, and others none? If the whole body

were an eye where were the hearing, if an eare where were the smelling? so, were all contracted into one, what would be∣come of the rest? No, rather let every one be thankefull un∣to God for that hee hath, then envious for that hee hath not; God hath store enough for all, let us not envie one ano∣ther; if one pull downe the house as fast as another builds, what will become of the building? rather let al•• build and then a goodly structure will be sooner finished,

Quod Sapimus conjungat Amor, quod vivimus uno Conspiret studio; Nil dissociabile firmum.

Nature it selfe peacheth us to make much of union, with∣out which all falls to ruine; See we how the dayes part equal•• stakes with the nights, and if one have a greater share now, it allowes the other it afterwards.

The Sun not all doth claime, but Moone and Starres have shares, And greater lights neede not envie the lesser theirs: The Land not all doth keepe but gives the Sea its bounds, Hills have their tops, and let the valleyes have their rounds.

The sweet combination of Elements and inanimat bo∣dies may teach us concord; Fire and water, earth and ayre how soever opposite in themselves in both qualities, yet in mixtion and composition sweetly conjoyne together, one not destu••bing but qualifying the excesse of the other, as if they without eyther sense or reason meant to teach men both: But above all, Piety and Religion enjoynes this, Are wee not all Christians, nay are wee not all Prote∣stants (or at least should be) all professing one and the same faith, the same Baptisme? have we not all one Father, one Redeemer, are wee not all members of one and the same holy Church? whence then are these discords amongst brethren, these detractions amongst friends? Ile speake

no more, least I seeme to teach those who should teach others. And this I speake not with any solace to my selfe, but with greefe of heart in the behalfe of such which are guilty of such breakings out, and make my prayers to God for them, that they may see the foulenesse of the offence, and be incited & stirred up unto repentance, and make satisfaction for the same, by calling in their sini∣ster and untrue reports, And as they render their owne repu∣tation, so to bee carefull of the reputation of others, and as they love their owne peace so to bee studious and ten∣der of the peace of others; If they shall doe this there will bee some satisfaction made (though full recompen∣sation for Calumnie can never bee made) and comfort in their acknowledgement, and repentance. If it can∣not bee by this effected, I will follow the precept of my Sa∣viour, and pray to God for t••em, and walke as well as I can through good reports, and bad reports, and comfort my selfe with the companie of my Saviour, of the Apostles, and of all Gods Saints, who were not exempted from the lash of the tongue, commending my cause to God (and to the equall judgement and censure of a Reader not partiall) to whose mercy likewise I commend them, and so desiring with S^t. Paul that we may all keepe the unity of the Spirit in the band of peace, this ever shall be my wish

Hoc habeat concordia signum Vt quos una fides, jungat & vnus Amor.