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 performe the Golden Rule, or to finde a Proportionall Number unto another Number, as two other Numbers are in proportion amongst themselves.

THis Rule of all other is the most excellent and the most generall, as well in Mathe∣maticall Calculations, as in Arithme∣ticall Computations, and therefore may not unfitly be so called, and the In∣strumentall operation is rather more fa∣cile in this Rule, than in Multiplication or Division; hence it is that I have dispo∣sed it in the front of the worke, because of expedition and facility: and the way of operation is thus:

Seeke the first number in the moveable,* 1.1 and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed.

Example 1

If the Interest of 100. li. be 8. li. in the yeare, what is the Interest of 65. li. for the same time.

Bring 100. in the moveable to 8. in the fixed,* 1.2 so right against 65. in the moveable is 5. 2. in the fixed, and so much is the Interest of 65. li. for a yeare at 8. li. for 100. li. per annum.

The Instrument not removed,* 1.3 you may at one instant right against any summe of money in the moveable, see the Interest thereof in the fixed: the reason of this is from the Definition of Logarithmes.

Proportionales Logarithmi aequales habent differentias.* 1.4 Necess•• est igitur proportionales Logarithmos in proport••one Lincari distantias aequales habere.

Example 2.

If a Troope of 50. Horse have for their pay 140. li. how much shall suffice to pay a Troope of 64. Horse.

* 1.5BRing 50. to 140 then right against 64. in the moveable is, 179. 2. in the fixed, the monthly pay of the said 64. horse. And there immediatly may you see the monthly charge of any number of Horse, for, the number of Horse given in the moveable, right against it, is their pay in the fixed.

Example 3

It is said that the proportion betweene the circumference of a Circle to his Diameter is ••. 22.

Bring therefore 7. in the moveable to 22. in the fixed, then immediatly at one instant may you have the Diameter or Circumference of any Circle, only by an ocular inspection: for right against the Diameter in the moveable, is the Circumfe∣rence in the fixed; or right against any Cir∣cumference in the fixed, is his Diameter in the moveable: Thus for the simple Rule.

Example 4

Further uses of the Golden Rule in ordinary service in propo••tionating of things.

Let FLX. represent the Perimeter of a Pentagonall Fort, and let the distance betweene the points of the Bastines, FL. be 926. foot, or KL the square side of a Building 470. foot, and the other dimentions, both of the Fort, and the Building ac∣cording to the here under inscribed Tables.

• The distance betweene the points of the Bulwarke. FL. 926.
• A perpendicular C R. 617.
• The Cottine A B 662.
• The side of the Fort D N 425.
• The gorge line A D. 119.
• The Flanke D E. 100.
• The line of defence D L. 700.
• The face of the B••stine E F. 264.
• The cap••tall li••e A F. 224.
• The distance from the Center to the Bastine A C. 564.
• From the Cottine to the Center C I. 456.
• The bredth of the Bulwark. G E. 310.

[illustration]

• The greatest square side of the Building K L. 470. foo••
• Q. A court within the middle of the Building.
• The distance betweene the middle of the Court and any out angle, as K A. 236.
• The least inner square of the Court E F. 200.
• Betweene any out corner of the Building, as RX. 180.
• 0. 0. 0. &c. a stone Gallery in bredth 36.

And so of other under roomes to other uses.

NOw admit another like Fort, or another like Building is to be erected, whose greatest distance betweene the aforesaid points of the Bastines, can be but 750. foot, or the greatest side of the peece of ground where the Building is to bee made, is but 400. foot, what shall the severall measures of this new Structure be, so that the Fort to the Fort, or the Building to the Building, in all parts be proportionall?

This is performed with much facility and expedition by this Grammelogia.* 1.6

For if you move the whole to the whole, viz. 926. to 750. or 470. to 400. right against the severall knowne measures in the moveable, you have the severall required measures in the fixed. I bring therefore 926. unto 750.

So right against

• 637
• 662
• 425
• 119
• 100
• 700
• 264
• 224
• 564
• 456
• 310

In the moveable is

• 515. 9.
• 536. 1.
• 344. 2.
• 96. 4.
• 81. 0.
• 566. 9.
• 213. 8.
• 181. 4.
• 456. 8.
• 369. 3.
• 251. 1.

In the fixed So right against

• 236
• 200
• 180
• 36

in the move∣able is

• 285. 9.
• 170. 2.
• 153.1.
• 8

in the fixed.

These numbers found out by the ordinary way of Arith∣meticke may trouble a nimble Arithmetician a whole houre or more, and therein subject to much error, but others 6. or 8. houres at the least, if not more; but by this Grammelogia, they are found out in lesse time than halfe a quarter of an houre: for so quicke is its operation in any question, to him that hath the way of working by it, that it gives the Answer before a man can distinctly write downe the numbers propo∣ed in the question.

Further uses of the Golden Rule, in matters of combination of Numbers, how to part a number into parts, as another number is already parted.

• LEt A. B. C. D. E. be five men which adventure money in a Plantation or otherwise: A. adventures 84. li. B 72. li. C 48. li. D. 54. E 42. li. by which in the returne is gotten 50. li. how much shall A. B. C. D. and E. have, according to their severall disbursments.
• Or admit F. borroweth of A. 84. li. of B. 72. li. of C. 48. li. of D. 54. li. and of E. 42. li. F. dies, and his whole estate is worth but 50. li. how much shall every Creditor have of this 50. li. according to his money lent.
• Or suppose A. B. C. D. E. were five severall metals, alotted to make a Statue, Vessell, Bell, &c. A Gold, B Silver, C Co∣per, D Latten, and E Tin; now when the Metals were melt and cast, there was left a peece which weighed 50. li. how much Gold, Silver, Coper, Latten, and Tin doth it containe, that so the worth of that peece may be knowne.
• Or if there were 5. Companies, or 5. Captaines, A. B. C. D. E. who expect their Pay, to A was owing for his service 84. li. to B. 72. li. to C. 48. li. to D. 54. li. and to E. 42. li. Now to keepe them from mutiny, the Generall sends them 50. li. to be parted amongst them proportionally according to each others dues, what shall A. B. C. D. E. have?
• Or admit A. B. C. D. and E. should load a ship of 300. tuns, A layes in 84 tuns, B. 72. C. 48. D. 54. and E. 42. tuns; in the voyage by reason of tempest, for safegard of their lives and Ship, there was cast over boord 50. tuns of the loading, how much shall A beare of the losse, as also B. C. D. and E.
• Further, in a Shire there is to be raised of 5. men, A. B. C. D. and E. 50. li. proportionally according to their estates; A is worth yearely 84. li. B. 72. li. C. 48 li. D. 54. li. and E. 42. li. how much shall each one pay, &c.

THus I might infinitely dilate my selfe upon one subject, tending to admirable uses, I onely in this glance by things, making but way to the occasions: The resolution of which, and all othets of this kinde, is drawne from this en∣suing Axiome.* 1.7

There is such proportion betweene any whole, and his parts, as betweene the like whole, either greater or lesser, and his parts: or betweene the parts and the parts, as betweene the whole and the whole.

So in the first example,* 1.8 Adde the money of A. B. C. D. and E. together makes 300. li. this is the whole, the parts are the former: now 50. li. is another whole number, which must be broken into parts proportionall to the former; and this differeth nothing in the operation from that of the last, in proportionating the Fort to the Fort, or the Building to the Building: for such proportion as 300. li. the whole money disbursed hath unto 50. li. the whole money gotten, so shall A 84. have to his part, and so of any other.

Bring therefore 300. in the moveable unto 50. in the fixed,* 1.9 so right against any particular part in the moveable is his part proportionall in the fixed, as there apparantly is seene, and from thence they are taken and placed in a Table, as here under appeares.

As 300. to 50. so

• A. 84.
• B. 72.
• C. 48.
• D. 54.
• E. 42.

to

• 14.
• 12.
• 8.
• 9.
• 7.

More uses upon the Golden Rule, in the division of Lines.

[Propositio. 1] TO finde a Line that shall keepe any proportion assigned unto another line given.

* 1.10As, let a Line be found which shall keepe proportion to the line A. as 3. to 5.

[illustration]

* 1.11Measure the line A by a scale of equall parts, then bring 3. unto 5. so against the measure of the line A in the moveable, you have the measure of the line requi∣red in the fixed, viz. B. so the lines A and B are in proportion as 3. to 5. &c.

[Propositio. 2] To diuide a Line into any number of equall parts.

* 1.12Let it be required to divide the Line A into 23. parts: first, by a seale of equall parts measure

[illustration]

the Line A,* 1.13 which admit to bee 51. parts, bring then 23. in the moveable unto 51. in the fixed. So right against 1. 5. 10. 15. 20. in the moveable, is 2. and 2. 10. 11. and 1. 10. 22. and 2. 10. 33. and 2. 10. 44. and 3. 10. in the fixed: if these numbers be taken from the same scale, and applied to the line A, it will be divided in the points of 1. 5. 10. 15. and 20. then may those parts be ea∣sily sub-divided.

[Propositio. 3] To divide a Line in such sort or proportion as another Line is already d vided.

* 1.14Let the Line B. C. bee divided in the points, D. E. F. G. and H. as the Line R. is divided.

* 1.15Measure the Line R. 58. and his divisions R. 12. R 15 R. 20; R. 30. R. 50. then let BC. be measured, which admit it con∣taine 37. parts, bring 51. unto 37. so against the parts of R in the moveable, you have the parts of B C. in the fixed, viz. B D. B E. B F. B G. and B H.

[Propositio. 4] To finde a line in continuall proportion unto two given lines, or a proportionall line to 3. lines, it differeth nothing from that of Numbers, and therefore wrought accordingly.

Notes

• * 1.1

  Constructio.
• * 1.2

  Constructio.
• * 1.3

  ☜
• * 1.4

  Definitio.
• * 1.5

  Constructio.
• * 1.6

  Constructio.
• * 1.7

  Axiome.
• * 1.8

  Declaratio.
• * 1.9

  Constructio.
• * 1.10

  Declaratio.
• * 1.11

  Constructio.
• * 1.12

  Declaratio.
• * 1.13

  Constructio.
• * 1.14

  Declaratio.
• * 1.15

  Constructio.