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.

How to operate, in the finding of Proportionalls by my Logarythmall Projection of Circles inlarged, eyther by a mooveable and fixed Circle, or by a single Projection, with an Index at the Peripheria, or Center.

THe way of operation is drawne from the nature of Proportionall Logarythmes, that as they keepe equall differences, so in a lineary or Circular Instrumentall projection of these Logarythmes, Proportionall numbers shall alwayes have equall distances, which as a fundamentall ground may serve to the more learned both for a full demonstration, and direction in operation.

But to make things more obvious, & to remoove such scruples as may arise in working by this Projection; the numbring ••f the Circles espe∣cially is to be considered, that is either by augm••••••ation or diminu∣tion, in continuation or discontinuation, and th•••• ••ath relation to the line of Conjunction E. T. which sheweth the breaches of the Circle, or the uniting or continuing of the parts: which multiplicitie of Circles, must be conceived to be but the parts of one Circle (as before amply in the Epistle to the Reader was specified touching this projection) and so continued or discontinued, by ascending or descending on this or that side of the line of coniunction, as by the succession of the Graduati∣ons, or divisions in those Circles is most evident and conspicuous: this well premised:

Const••uctio. Bring the first number in the mooveable, to the se∣cond n••mber in the fixed, and marke the severall revolutions or Cir∣cles betweene them ascending or descending; for then the fourth Pro∣portionall is had on the fixed, right against the third number in the mooveable by the same number of revolutions or Circles ascending or descending, as was betweene the first and second numbers.

So if the line of coniunction on the mooveable, be on the right or left side on the line of Coniunction on the fixed, and the first and second numbers be betweene them, and also the third, or all these three num∣bers be not betweene them, the fourth number or proportionall is had without any consideration, but onely by the same number of Circles, as was betweene the first and second numbers. But if the third num∣ber be on the otherside of the line of Coniunction, and that the Propor∣tionalls

did augment, or diminish, the same number of Circles or revolutions accounted ascending or descending from the third number, will likewise shew the fourth proportionall required.

Or without considering the absolute revolutions, you may operate by the difference of Circles noted on the single Index thus.

Bring the first number in the mooveable unto the second number in the fixed, and marke the difference of Circles, eyther ascending or descending betweene them; then whensoever in operation the sayd first and second number, and also the third number are neyther of them betweene the lines of Coniunctions, or all of them are betweene them, then the fourth proportionall is had in the fixed right against the third number in the mooveable, by the same difference of Circles ascending or descending, as was betweene the first and se∣cond numbers.

But if the line of conjunction on the moveable, bee on the right side of the line of con∣junction on the fixed, and the first number bee in a lower Circle then the second and

• bee not be∣tweene the lines of Con∣junctions, but the third,
• be betweene the lines of conjunctions & not the third

then the fourth proportio¦nall is in one Circle

• ...more
• ...lesse

then the diffe∣rence of Cir∣cles betweene the first and se∣cond numbers. But if the first number bee in a higher Cir∣cle then the se∣cond, then the difference more will bee lesse, and that which is lesse more.

Contrarily, if the line of Conjunction on the mooveable bee on the left side of the line of Conjunction on the fixed, the operation is the con∣verse of the former.

The like may be observed in operation by the single Pro∣jection inlarged, with a double Index to move on the Peri∣pheria, or Center of a Circle, if in stead of moving one num∣ber to another, you extend the feete of the Index, as followeth.

An Example of the operation upon the Projection of the Circles of my Ring inlarged, according to the conclu∣sion of my first Booke, in a Scheme or Instrument where the Circles of Numbers, Sines, and Tangents are decuplated, the diame∣ter being but 18. Inches onely.

SVppose the Sunne being in the Tropicke, and his greatest Amplitude were required in the Latitude of 66. gr. 29. m. The common Rule to operate which, is thus delivered.

As the Sine complement of the Latitude, is unto the Sine of the Suns declination proposed, so is the Sine of 90. gr. unto the sine of the Suns Amplitude required; the Diagramme and Demonstration of which, see pag 57. and the Instrumentall operation may be ••hus.

Constructio. Place one of the edges of the Index unto the sine of 23. gr. 31. m. (the Complement of the Latitude proposed) and extend the other edge unto the Sine of 23. gr. 30. m. (the Suns greatest declina∣tion) then moove the edge of the first Index unto the Sine of 90. gr. so the edge of the second Index shall give the fourth proportionall sought for.

But because the answer in this Proposition falls neare the sine of 90. gr. (where the degrees are small, and the graduations close) the answer is not so exactly discerned in the minuts as if it were larger. To sup∣ply which, or such like as may fall out in Practise, I have continued the Sines of the Proiection unto two severall revolutions, the one begin∣ning at 77. gr. 45. m. 6. s. and ends at 90. gr. (being the last revolution of the decuplation of the former, or the hundred part of that Proiecti∣on) the other beginning at 86. gr. 6. m. 48. s. and ends at 90. gr. (being the last of a ••ernary of decuplated revolutions, or the thousand part of tha•• Proiection) and may bee thus used.

Lay the edge of the Ind••x upon the former 23. gr. 31. m. and marke what equall parts in the Circle of equall parts are intersected there∣by, which will be 0099. Place also the edge of the Index upon 23. gr. 30. m. which will cut in the Circle of equall parts 0069. Then take ten times the distance of these two numbers in the Circle of equall parts be∣tweene the feete of the Index by decuplation, that is, place one foote of the Index upon 99. and extend the other foote unto 69. so the edge of the first foote being placed on the sine of 90. gr. the edge of the second

foote will point out accuratly 87. gr. 54. m. and such is the Amplitude in the Latitude of 66. gr. 29. m. the Sun being in the Tropicke aforesaid.

Againe, admit that in the Latitude of 51. gr. 32. m. the Suns place were required by knowledge of his Meridionall Altitude, which suppose was found to be 61. gr. 57. m ⅙. Now the common Axiome according to the Diagram of Pag. 56. grounded on Mathematicall doctrine to operate which is: As the sine of the Suns greatest declination, viz. 23. gr. 30. m. is unto the sine of 90. gr. so is the sine of the Suns Declination given viz. 23. gr. 29. m ⅙ unto the sine of the answer. Therefore In∣strumentally open the Index unto the first two tearmes proposed, and place the edge of the first foot upon the third number, so the edge of the second foote shall give the fourth proportionall required.

But for as much also as the answer in this doth fall amongst the small divisions neare the sine of 90. gr. in the decuplated Proiection, we may supply that contraction according to the former, but truer by the lar∣gest augmented Circle of Sines, to wit, that which is the thousand part of its Proiection, thus; Lay one of the edges of the Index upon the sine of 23 gr. 30. m. and marke the number of equall parts which it cuts in the Circle of equall parts, viz. 70. lay also the edge of the Index upon 23. gr. 29. m. ⅙. and it cuts likewise in that Circle 64. then take a hundred times the distance of these two numbers in the Circle of equall parts betweene the feete of the Index, that is, place one of the edges of the Index upon 70. m the Circle of equall parts by Centuplation, and ex∣tend the other edge unto 64 (for the Decuplating, and Centuplating of the equall parts, doe naturally without trouble represent themselves in this projection) so the edge of the first Index being placed at the sine of 90. gr. the other edge will accuratly point out 89. gr. 5. m. the Suns place at the time observed, and so in like manner for other operations.

Nam quam in minoribus circulis aequalium partium distantiam pro∣portionales habent: eandem in circulis adauctis proportionales re∣tinebunt.

Thus I have here now produced to a publike veiw my Proiection of Logarythmes inlarged by way of decuplating, Centuplating, &c. of the Circles, as I promised at the Conclusion of the first publishing of this In∣vention, to the world, and have in some measure shewed the Accurate working of Trigonometry by it, neare the Sine of 90. gr. where difficulty seemes to be; for being thus inlarged the greatnesse of a degree be∣tweene the sine of 89. gr. and the sine of 90 gr. is more then 4. Inches, and if I should have inlarged the Circle of Tangents according to the Sines, the capacity of a minute at 45. gr. would bee more than 8. Inches

which inlargement amongst the degrees which falls neare the Sine of 90. gr. doth operate as true (according to the former Diameter but if 18. Inches) as if M^r. Gunters excellent Lines were extended or projected unto 4000. foote. And if I should frame a Ring as is specified in the conclusion of my first publication of this Invention, being of two yards diameter, and apply it to Astronomicall Calculations, no doubt I might shew a way (or others may easily) to compendiate many ope∣rations therein; and sufficiently cleare my intentions then delivered touching the Prostaphereses of the motions; though some one in con∣tempt of my good indevours divulged after it came to the worlds view, it could not bee done, nor possibly a minute expressed at the sine of 90 gr. (as I have now produced it) thereby endeavoring to annihilate my labours and to spread an unsavory rumor, which might seeme to argue not onely his ignorance of my intentions, but also of the manner of extending and inlarging of that invention (though now given out that I had that invention from him:) But touching such appli∣cations to prove my assertion hereafter, as God shall give life, & ability of health: and let a further time bring them to maturitie, that my Iea∣lous opposite may bee no more mistaken with a suspected untimely birth. I confesse these single Circles before, were something untimely, in regard of these which are of fuller growth, and yet may have further application, without wayting his time to perfect them.