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 courteous and benevolent Rea∣der that affects this instrumentall practise of the Logarythmes projected circular Intituled Grammelogia, or the Mathematicall Ring.

AS that Honorable Lord Nepeir (the first inventer of Loga∣rythmes) did call his Rods by reason of their facility in operation Rabdologia, the speech of Rods, and his numbers, Logarythmes, the speech of numbers: So this Instrumentall Projection as springing from that invention, I called Grammelogia, the speech of lines; for being so projected as they are, these lines graduated, doe promptly teach one what to speake in proportionall operations; which invention of projection of Logarythmes Circular, I promised to enlarge at the end of my first publication, with this intituled Grammelogia, or the Mathematicall Ring, Printed, Anno. 1630. notwithstanding it was there sufficiently perspicuous, how it might bee augmented unto any magnitude assigned, even to operate unto minuts, and seconds in Trigonometrie, and to finde Rootes, and proportionall numbers in com∣mon Arithmeticke unto sixe or more places: a diagramme of which projection I now here deliver, as fully sufficient to shadow out to the more learned the quintessence of this Logarythmall projection in Cir∣cles unto 6. 10. 40. 100. or 1000. yea, to as great a capacity as one de∣sires, (a project by Instrument never yet produced, though long desired) which plurality of Circles in this projection, must be conceived to bee the parts of Circles, and so there may bee a quadruplicitie of them,

one of which Quadruplicities, may co••••••ine the Loga••••th••es ••f ••••••∣bers, another may comprehend the L••••arythmes of sines and the other two may bee for the inserting of the Logarythmes of Tangents: so in this Scheme here delivered noted with the letter B. there are 16. Circles in view, but 4. in effect, each of which 4. must be conceived to be broken into parts is before: in the first of which Quadruplicities of Circles, there are graduated the Logarythmes of ••••mbers, (as be∣fore) from 1. unto 100000. noted with the letter N. and so may bee properly called the Circle of numbers: in the next quadruplicitie of Circles there are graduated the Logarythmes of sines, from 34. m. 22. se. unto 90. gr. noted with the letter S. containing two revolutions, that is, two perf••ct Circles, & therefore is divided & figured double, beginning his first revolution at 34. m. 22. se. and ending at 5. gr. 44. m. 21. se. which are graduated upon the inner edge of the Circle the second re∣volution begins at 5 gr. 45. m. 21. se. and ends at 90. gr. which are gra∣duated and figured on the outward edge of the Circle, which gradu∣ations and divisions of sines, may be therefore called the Circle of sines. Lastly, the next two quadruplicities of Circles which are noted with the letters T. T. do containe foure revolutions or Circles, because also of their double divisions, in which are inserted the Logarythmes of Tangents, from 34. m. 22. se. unto 89. gr. 25. m. 38. se. beginning the first revolution at 34. m. 22. se. and ending at 5. gr. 42. m. 38 se. which are divided and figured upon the outer edge of the Circle; the second revolution begins at 5. gr. 42. m. 38. se. and ends at 45. gr. and are graduated and figured also upon the outerside of the Circle. The third revolution begins at 45. gr. and ends at 84. gr. 17. m. 21. se. and are graduated and figured upon the innerside of the Circle. Lastly, the fourth revolution begins at 84. gr. 17. m. 21. se. and ends at 89. gr. 25. m. 38. se. all which graduations of Tangents may be called in like manner the Circle of Tangents. And as that famous of memory (if not most injuriously late made infamous) and worthy Mathematiti∣an, Mr. Gunter, the first that gave light to this Invention, did call his lines the lines of Numbers, Sines, and Tangents, or the li••es of propor∣tions; so this whole projection as adherent unto it, may not unfitly be called the Circles of Numbers, Sines, and Tangents, or in respect of operation (as some others lately called the Circles of my King singly projected on a Plate or plaine) the Circles of proportion, in which may be noted, that if this projection be made of 10. Circles, then may there be a quadruplicitie of them, which in all make 40. Circles, one quadruplicitie of which may serve for the inserting the Logarythmes

of numbers, another for the Logarythmes of Sines, and the other two for the Logarythmes of Tangents. If the projection be of 50. Circles, then the whole projection will be of 200. Circles, whose graduations, beginnings, and terminations in each whole revolution is the same with the former; and as there is a conformity in the Projection by a greater number of Circles, as there is by a lesser; so is there the same facilitie and agreement in operation by many Circles, as by few, and the way how I have delivered in this Tractate, which is either by motion in a double projection, or in a single projection, by helpe of a thred & lead, or a single or double Index at the Center, or periphe∣ria: upon which I deliver severall wayes in this following Treatise. How it may please some mens affections I know not, my intentions and desires are free. Now since the publication of the first kinde of Logarythmes Projection, Anno 1630. or the originall of this enlar∣ged, it hath pleased many about this City and Kingdome to take liking thereunto, some contenting themselves with the double pro∣jection, with a moveable and fixed Circle, some with the single ha∣ving an Index at the Center: but generally the most part have beene by some invited if not forced to that which carries, as they say, with it the aplause and vote of men by a comparative attribution delivered by the assumed Author of that with an Index on a Plate, that the way on the plate in a single Projection with an Index at the Center, is a bet∣ter way than that of my Ring, or that of a double Projection on a Plaine: The authority of whose words to some ignorant mechanick composors of that Instrument, was a sufficient motive ever since to crowne his words with a divulged rumour out of their borrowed knowledge to maintaine his assertion, to put off their commodities, howsoever if the saddle (as the proverbe is) were put on his right place, this vote and attribution belongs to another, who fitted the In∣strument so as it is now used, & yet modestly would not appeare in it, and not our supposed Author, which assumption of fitting of it so with an Index on a Plate, had beene enough if not too much, without such a divulgation of endeavouring what in him lyeth, and in others, to annihilate and beate downe the way which I write upon, and to glory in the raising up of his supposed owne: thereby not onely possessing men with an untruth, but making me also ignorant in my choyse, that I should give unto the world the weakest and imperfectest part of the projection of Logarythmes, and leave the best for another to write up∣on: Of which single Projection with an Index at the Center, had I first writ upon and left the other way, which is the double Projection

for some other to write upon: then might he have used not indirect∣ly that comparative aspertion of better. But before I writ of the Naturall sympathy of this projection, I was not unadvised which pro∣jection to present unto the King, and to the publike view of the world first, but considered intentively with my selfe the excellency of both wayes, and the more copious performance of the one, in re∣spect of the other; And why I delivered it first in a Ring, was, for the aptnesse and gentile-forme (as I may call it) it naturally might be cast into. Secondly, for the excellent harmony, facility and expe∣dition that the Logarythmes so projected did afford: having no secon∣dary assistance to helpe it in operation, but the motion of the Circle it selfe, for there was nothing to do but to move one number to ano∣ther in a proportion assigned, either in a double Projection in single Circles, or the projection inlarged; and instantly there was presented all other numbers in the same proportion; By an Index on a Plate in a single Proiection it was grosse and course for the forme (in respect of a Ring) and for operation there must bee besides extending the feet of the Index to the members, upon every severall question a new search of numbers with a new motion (which extending of the feet of the Index, was the same with M^r. Gunters Invention of his Ruler and no new invention) Besides if the single Projection bee inlarged there doth necessarily adhere unto it sundry & manifold observations in the way of operation by it, which cannot be avoided, which to a lear∣ner at the first seemes not a little harsh & difficult; all which the Ring or the way of the Ring on a plate, in the double Proiectiō inlarged, doth naturally avoyd, and not only caries a facility in its operation, but re∣taines in it also a speciall advantage in its performance, once rectifi∣ed, for the eye and the hand may worke together, and what the eye finds in proportion, the pen may presently expresse in writing with∣out a second trouble to search out another number as before, and then to bring the edge of the Index to it. But some envious detractors would not admit of this forme and facility (though perhaps the suc∣ceeding times may) eyther as before to disanull the worke, or for the difficultie that was found in an unexpert workeman in the true com∣posing and making of the worke, for if the Circles on the Ring, or double Proiection on a plaine being not exactly composed and gradua∣ted may cause some small error in operation (which is onely from an excentricke motion) the single Proiection hath not onely the same defect, but also a second to helpe it, to wit the Index, for by how much the legs of it are long and the Instrument large, by so much the more

is it subject to errour, which is from a continued augmentation of an error in the fitting of it to a lesser Circle which hath reference to a greater. But to passe by the errors that are subject to the best kinde of Instruments that can be made, let us a little examine the Authors & others comparison of Better, why the way of the Index in a single Proiection is better then a mooveable and fixed Circle, which I con∣ceive to have reference unto foure generalls about the Instrument. First, either in the forme of the Instrument; Secondly, in the orde∣ring of the Circles thereon; Thirdly, the expedition that is found in the practise thereof; And fourthly and lastly, the copiousnesse of the uses of the Instrument: and other c••uses I conceive not, why this sin∣gle Projection with an Index is better then a double, except it be in the magnitude that is now usually made, or for the price of the In∣strument: in the first there may bee an equall extendure of magni∣tudes unto both Instruments, and so as a thing common unto them, and no wise different. And for the price it may bee made as cheape, if not cheaper hereafter as I shall order it for these that aflect them: Now in the first place as touching the forme, upon that I have spoken somewhat alre••dy, (as afore said) and may be sufficient: as for the second generall touching that of the ordering of the Circles on this double projection, to have one Circle mooveable and one Circle fixed, that is agreeable to the projection and dividing of these Circles in the first direction following, according to the great scheme in the Booke noted with the letter A, for if the Circle of numbers noted with N, N, be cut through, and the Center of that Circular plaine be fastned, so that it may move upon the Center of the other Circle, it shall fully represent the projection of my Ring upon a Plaine, to which may bee placed a small single Index as in the scheme of the title page B, to helpe the eye for the finding of opposite num∣bers, these Circles of the mooveable, or fixed Circles on the plaine are inserted on both sides of a Ring as it is specified at the end of the di∣viding of these Circles. Now to have all the Circles placed upon one side of the Ring (as is according to the second direction of making the Ring) were to leave the other side naked, without one would patch and peece some other thing on the other side, therefore to a∣voyde mixture, part of the projection (as an ornament) is placed on the other side, by which occasion the whole projection at once is not visible to the eye, as it would be in the second way in accommodating the Circles into a Ring, as is in the double projection on a Plaine be∣fore mentioned, noted with A. But perhaps it may be objected that

the Circles are easily continued on a Plaine, and the Index being at the Center the edge of it, doth accuratly cut each Circle in the pro∣portionalls, which intersections in a Ring are defective and difficult to finde by the eye alone. To all which I answer, that the Circles in a Ring are as easy to be continued as on a Plate, allowing the moove∣able and fixed Circle a sufficient breadth, and here by the way I would have the Reader to understand that the Circle of Tangents being pro∣jected from 1. gr. unto 45. gr. is sufficient for operation (these degrees being their complements to 90. gr. but for greater expedition in wor∣king they may be continued as is seene in the great scheme A, which continuation I learned not from another (as may bee suspected by some) seeing I now published it after another, but long before that publication I instructed sundry persons upon that continuation by way of facility. As for the second clause in such Circles which are not upon the edge of the mooveable and fixed Circle, where the eye seemes to bee troubled to point out some opposite numbers, a small edge of metall may easily supply that (as many use to doe) but the graduations being so nere the edge of the Circle, the proportionalls are sufficiently given by the eye alone without such an edge. Now if in these respects the single projection with an Index, is better then that of a mooveable and fixed Circle being easily supplyed as afore∣said, it is but a poore one, in common sense. But if the way with an Index on a single Projection bee not better then that of a double for the former respects, then it may bee in the third generall to wit the Instrumentall expedition: in which there needes little declara∣tion to prove the double Projection to have a greater expedition then the single projection with an Index, seeing it appeares so obvious, that what can be quicker; then having mooved one number to another in proportion assigned, that all other numbers are opposite one to ano∣ther in the like projection (which a single Index doth point out easily to the eye as before.) By the Index in a single projection there is first putting the one foot to one number, and extending the other foot to another number, then must the eye have reference to one of the feete that it fall upon his third number, and afterward to looke for the second foote for the fourth number, and so to move it to another number, and still to have a double respect with the eye as before in every new operation: in this regard also I see not why the way of an Index in a single Projection is better then that which I have delive∣red in a double projection. But to passe by all the former generalls as triviall and of small consequence, let us weigh seriously things more

materiall touching both wayes of these Instruments. If the said single projection on a plate with an Index at the Center bee not better in respect of its expedition in operation, then must it necessarily bee (to prove the Authors assertion) in the Instruments fourth generall, to wit, in its copious performance, which I hold eyther to be in the gene∣rall, or particular use; In the generall I considered the finding of proportionalls, and thats agreeable to the way of operation in either Instruments as is afore specified; In the particular I regard also what propositions offer themselves to the eye, eyther by motion, or without motion: by motion it is impossible for the single projection without an Index being opened at pleasure to give any more then one kind of proportionalls, the Ring, or a mooveable and fixed Circle on a Plaine, scorning as it were such lamenesse, or such an injurious tie from its naturall propertie, sheweth by motion infinite operations in various proportionalls, even through the whole body of the practicall part of Mathematicall Art, which would be too copious for me to de∣clare, or for the Readers patience to peruse, onely some common uses by such motion I will deliver, somewhat to prove my assertion, that the single proiection with an Index, is not better then that with a mooveable and fixed Circle, therefore.

1. First, the moveable being moved about at pleasure, as 1. in the moveable passeth by any multiplier in the fixed, so doth any multipli∣cand in the moveable, point out its product in the fixed, or contrarily, as any divisor in the moveable doth passe by, 1. in the fixed, so doth any dividend in the moveable point out his quotient in the fixed, so as 12. (the months in a yeare) or 52. (the weekes in a yeare) or as 365. (the dayes in a yeare) in the moveable in motion doth passe by 1. in the fixed, so any somme of money in the moveable, doth point out its monthly, weekely, or dayly expences in the fixed.

2. Secondly, as 7. in the moveable doth passe by 22. in the fixed (Ar∣chimedes Proportion betweene the diameter of a Circle and its Cir∣cumference,) so doth any diameter in the moveable, point out its Cir∣cumference in the fixed, vel contra.

3. Thirdly, as a hundred waight of any commodity in the moveable, (or any other waight or measure) passeth by its price under 100. pound in the fixed, so right against 1. in the moveable, is the price of a pound waight of that commodity amongst the decimals in the fixed vel contra.

4. Fourthly, the moveable being moved about at pleasure, the Interest of all summes of money according to any rate in the hundred is given;

for as 100, in the moveable passeth by its interest in the fixed, so eve∣ry summe of money in the moveable, doth point out its interest in the fixed, vel contra.

5. Fiftly, as 1. in the moveable passeth by any sum of money in the fixed, so any number of yeares in the Circle of yeares, doth point out the amount of that money in the fixed, according to the terme of yeares that th•• money was forborne.

6. Sixtly; as the measure of a side, of any dimension, of a Building, of a Fortification, of a whole mixture, or the weight of it, &c. in the moveable passeth by a greater, or lesser measure, or waight in the fixed (in homogeniall things) so the measures of the parts of any of these wholes in the moveable will point out in the fixed the proportionall parts of any other whole by way of augmentation, or diminution.

7. Seaventhly, as 1. in the moveable passeth by the square of the side of any of the tenne regular Plaines, so doth each plaine note in the moveable point out right against it, its Area in the fixed; and as any kinde of measure to the Pole in the moveable, passeth by its quanti∣ty in the fixed, so doth any other kinde of Pole point out its quantity or Area, being measured by that Pole, &c. and whatsoever may be at∣tributed to the use of this Circle of numbers may be given by mo∣tion.

Further, if we consider the Circle of Sines and Tangents conjoyned with the Circle of Numbers in operation, or the Sines with themselves or joyned with the Tangents, then by motion you have the sides and Angles of infinite plaine and Sphericall Triangles for practicall uses, either in Geometrie, Astronomie, Navigation, Fortification, Horot ogo∣graphie, Geographie, &c.

1. First, as the sine of 90. passeth by 60. in the fixed amongst the num∣bers, so the sine complement of any degree in the mooveable will point out the miles answerable to any degree of Longitude in the Latitude; and as the said 90. passeth by the Tropicall point, in the fixed, so the sine of any degree of the Sunnes distance from the Equinoctiall points will point out the sine of the sunnes declination answerable to that distance.

2. Secondly, as the sine of any Rumbe in the mooveable from the East or West, sayled upon, passeth by the measure of a degree in leagues or miles in the fixed, so 1. in the mooveable pointeth out in the fixed the number of miles, or leagues to raise or depresse the pole a degree in that Latitude.

3. Thirdly, as the sine of any Latitude in the mooveables passeth by the

sine of the Tropicall point in the fixed, so the sine of the Sunnes distance from the Equinoctiall points in the moove∣able, that passeth by the sine of 90. in the fixed, doth point out the sine of the Sunnes greatest degree of the distance from the Equinoctiall points that the Sunne will bee due East in that Latitude.

4. Fourthly, as the sine of 90. in t••e mooveable passeth by the sine Complement of any Latitude in the fixed, so right against the sine Complement of all Declining plaines in that Latitude in the mooveable, are the sines of the degrees of the stiles heights in Horologographie agreeable to these de∣clining plaines in the fixed.

5. Fiftly, as the sine Complement of any Latitude in the mooveble passes by the Tropicall point in the fixed, so the sine of the Suns distance from eyther of the Equinoctionall points, will point out right against them the sines of the Suns Amplitude.

6. Sixtly, as the Tangent complement of any Latitude in the mooveable passeth by the sine of 90, in the fixed, so the Tangent of the Tropicall point in the mooveable, doth point cut in the fixed the sine of the greatest difference of ascention for that Latitude

7. Seventhly, as the sine of the Suns position at his setting or rising, or the sine of the houre from 6. at that instant in the mooveable passeth by the sine of 90. so the sine of the Suns declination in the former, & the Tangent of that decli∣nation in the latter, will point out the sine of the height of the Equinoctiall in the former, but the Tangent of the same in the latter.

In this nature you have infinit operations performed by motion in this double Projection of a mooveable and fixed Circle; which by a single Proiection with an Index cannot as before possibly be performed, therefore if in this regard the way of the Index on a single Proiection bee not better but is farre inferiour to that of a mooveable and fixed Cir∣cle; to prove the Authors and others divulged asser∣tion, then must it be better in the last clause, which was the Instrumentall performance without motion, in which the single projection with an Index, comes very short of other Instruments which by a single inspection of the eye shewes many pleasant, and usefull propositions. But this none, or very few at all, as onely the Logarith∣mes

of numbers, the naturall sines and the Tangents of the Logarythmall, &c. But the double projection with a mooveable and fixed Circle doth not onely shew that, but being at any position carries with it such an excellencie that it assumes unto it selfe a prehemenencie above any Instrument never yet produced in regard of its copious use, & manifold performances which it affords without motion as by an inspection of the eye onely: a touch of w^ch I will unfould & unvaile which never yet came to a publik view.

1. First, the Instrument lying upon a Table open to the eye and being at any position, marke what numbers in the mooveable and fixed are opposite one unto another, accor∣ding to which proportion there is represented infinite o∣ther proportionalls, in the same proportion, for one number is opposite to another through the whole Circle of num∣bers, sines, and Tangents, from which one might apply the proportionalls in numbers to the use of things, to expen∣ces, to proportions in Buildings, to fortifications, measura∣tions, but too great a prolix discovery would tyre the Rea∣der in that which he may easily from it apply hereafter un∣to himselfe.

2. Secondly, marke what number in the fixed, (in the Cir∣cle of numbers) is against any number of yeares in the mooveable, which suppose a Legacie to be payd for so many yeares to come or a summe of money forborne so long time; and it were to be sold for present money, right against 1. in the mooveable, is the worth in present of that Legacie or summe of money in the fixed.

3. Thirdly, in Horologographie, you have the distance of the houres in a poler, and Meridi••nall plaine, without operation, for marke what number in the Circle of numbers in the fixed is against the Tangent of 45. gr. in the moovea∣ble (which number in the fixed may be supposed the measure in inches, &c. of the stiles hight) so right against the Tan∣gent of the equall houres in the mooveable you have the houre distances in the Circle of Numbers in the fixed.

4. Fourthly, the houres of a Horizontall or verticall dyall, for some one Latitude or other is shewne, for marke what degree amongst the sines in the fixed (which represents the Latitude) is against the sine of 90. in the mooveable, so the Tangent of the equall houres in the mooveable, doth point

out the houre distances (for that Latitude) amongst the Tangents in the fixed, and these houres serve for a verticall Dyall in the complement of that Latitude.

5. Fiftly, note what number in the fixed in the Circle of Numbers, is against 1. in the mooveable (which suppose to be the given side of any of the ten regular figures) then a∣gainst the circumscribing and inscribing notes of these re∣gular figures in the mooveable is the Circles circumscribed and inscribed diameters of those regular figures: But if the said number in the fixed against 1. in the mooveable be ta∣ken for the square of the side of the diameter, of the side of any of the ten regular figures, then against the Regular notes in the mooveable, is the Area of these figures in the fixed.

6. Sixtly note what numbers in the Circle of numbers in the fixed, are against the regular figurative notes of equality in the moveable, such are the sides of those figures whose quantities are equall the one to the other: In like manner the numbers in the fixed against the notes of the Regular bodies in the mooveable representeth the sides of these bodies which have equall solidities the one unto ano∣ther.

7. Seventhly, you have infinite oblique angled plaine Tri∣angles represented, and such who have equall Altitudes but different Basis the sides of severall parallelograms, e∣quall unto one and the same square or the quantitie of a Triangle given in Acres, the perpendicular and Base is also given: For first the sines of the Angles on the mooveable will point out their sides of the Triangle amongst the num∣bers in the fixed, vel contra.

And secondly, marke what number in the mooveable in the Circle of numbers is against 1. in the fixed that suppose to be the Altitude of a Triangle, then the equall distances from 1. on both sides of it taken at pleasure in the fixed will point out in the mooveable the segments of the Basis of the Triangle, or the sides of a parallelograme equall unto the square made of the Triangles perpendicular. And thirdly, the quantitie of the Triangle in the fixed amongst the numbers doth point out the Basis of the Triangle in the mooveable, and that number in the fixed which is against AC, (in the mooveable) is the Triangles halfe

perpendicular according to the Area given.

8. Eightly, marke what number in the fixed, is against 1. in the mooveable, which suppose to be the side of any of the Regular bodies, then right against the solids inscribed notes in the mooveable, are their sphears Circumscribing dia∣meters, but if the said number in the fixed be supposed to be the semediameter of a spheare, the numbers in the fixed (in the Circle of numbers) against the solids circumscribed notes in the mooveable shewes the sides of these regular bodies that will circumscribe that spheare.

9. Ninthly, marke what number in the fixed amongst the Numbers is against 1. in the mooveable which may be sup∣posed the diameter of a Circle, the Axis of a spheare, the side of a plaine figure, or that of a solid body: so the numbers in the fixed in the Circle of numbers against the potentiall notes in the mooveable shall represent the diameter, Axis, or side of its homogeniall figure, or solid, according to the proportion of these potentiall notes in the moove∣able.

10. Tenthly, marke what numbers in the Circle of num∣bers in the fixed are against the notes of the regular figures, such shall bee their Areas, and the numbers in the fixed a∣gainst the regular Bodies convexities, such is their superfi∣ciall convexitie: and the number in the fixed against 1. in the mooveable, is the square of one of the sides of the regular figures, or the sides of one of these bodies, and what numbers in the fixed are against the notes of the solid bodies, such shall b•• the severall solidities, or contents of these regular bodies, and the number in the fixed against 1. in the mooveable is the Cube of the sides of these bodies.

Lastly, most courteous Reader, (not in any braving flourishes or branding any of the Nobilitie or Gentry with the attribute of jugling, against the simple modestie of the Author,) I have in some measure supported their honours in that particular in the Epistle at the end of this Booke: and that wee may say something more upon the excellencie of this Instrument without multiply∣ing of tautologized and needelesse prefixed graduall num∣bers, or Circuitions, if not Circumlocutions, in the naked truth of this Instrumentall proiection, according to its na∣turall propertie. The Roots of all square and cubicke

numbers without partition are given; and that by an in∣spection of the eye onely.

Thus I might have extended my selfe more copiously in the excellent use of this my mooveable and fixed Circle, and even from the Instrumentall position by an inspection of the eye onely without motion, compile a large Booke of its ample performance, but in that which I have delivered I have onely but scatteringly glanced upon things, as making way for many occasions, and as a motive to a further inqui∣ry: Its an ancient proverbe amongst us, good wine needs no Bush, but the wine must not be fast lockt up then, that none can come by it, if so it wants both bush and key, and to some such needelesse expressions might be avoided, the Instruments owne excellency will to the more learned ea∣sily present it selfe that which I have published: concerning it, I glory not in, but onely desire to satisfie those who would see the difference of both wayes, with, and without a mooveable Circle, & to let others know the truth of things which are conceated, and carryed away with opinion one∣ly, that the way of the Index on a single proiection is bet∣ter then the way of a mooveable and fixed Circle, which both in regard of expedition as also copiousnesse of the In∣strumentall use, by motion or without motion comes short of the other. What meanes the Authors divulgation then, that the way of the Index is better then the way of a moove∣able and fixed Circle, I know not, whose knowne skill in the whole Systeme of Mathematicall learning will easily free him from the suspition, that the way can be made, or the subject unvailed for him.

But I have now a little more made bold to unvaile the subiect for some, in the copious declaration of the excel∣lent use of this Logarythmall projection Circular by a moove∣able and fixed Circle, and also in its inlargement, which hitherto lay in obscurity, and as a generall benefit to those that affect the way of this Instrumentall practise. It were good that the divulgers would prove their aspertions, tou∣ching the word better, that others might participate sub∣stancially of their better way by the Instrumentall perfor∣mance, eyther by motion, or without motion, and not to al∣lure the world by a bare exhortation, unto the affection of the one Instrument, and by a dehortation to beate downe

the use of the other, which savours of too high a conceite of the one, and too great a detraction from the other: Too great & too loose an aspertion hath bin cast upon me about these things, which I never thought in the least title when I first writ upon this Invention, or my name so to come to the worlds rumor as it hath since the last publication of this Logarythmall projection Circular; howsoever, here is my comfort, the guiltlesnesse and innocency of my cause, which may teach me, and others carefulnesse here∣after, how and what we publish to the world, seeing there are such carpers, and maligners even of the most usefull and best things, yea, such busie bodies who marre that which others make, who scorne to have a second, knowing all things and admiring nothing but themselves, such who have stings like Bees, and Arrowes alwayes ready to shoot against these whom they dislike, such who while they will needs have many callings neglect their owne; sharpe wit∣tie cryticks, Diogenes like, snarling at others, and not looking home unto themselves, but by all meanes ende∣vouring to take away the mantle of peace, and rent the seamelesse coate of love and amitie. If things be not done well by others then they triumph and send forth their in∣vectives, if well, they professe it nothing, and cannot passe without their censure. To speake ill of a man upon know∣ledge shewes want of Charity; but to raise a scandall upon a bare supposition, & to act it in Print, argueth little huma∣nity, lesse Christianity: but enough of this if not too much, I am sure some have casted too much already, perhaps o∣thers hereafter may helpe to bare a share, for my owne part I desire no favour but the truth and equitie of my cause, and the due waighing of things with their reall cir∣cumstances. Ʋeritas non quaerit Angulos. I desire no shifting, or pretences, but if I have done others wrong let me suffer; If I have beene wronged by others let me have truth, and right done me, thats all I require. Who am

An ever well wisher to the truth and thee. R. D.