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.

Notions or principles touching the di∣sposing or ordering of the Numbers in the Golden Rule in their true places upon the Grammelogia, and the congruity of those Numbers one unto another.

NOte that in any question of the Golden Rule, there are three numbers to worke upon, whereof two of them are of one denomination, the one of them hath his answer, and the other doth require an answer, and those two numbers of like denominations must be alwaies accounted or sought out upon the moveable Circle.

[Example. 1] As if 30. li. doe rent 45. Acres of Land yearely, how much doth the yearely Rent of 84. Acres come to.

Here the denominations alike are 45. Acres and 84. Acres, 45. Acres hath his answer, 30. li. and 84. Acres requires his answer.

For the working of this and all others, Let the numbers in the moveable be brought to his answer in the fixed: that is, bring 45. to 30. so, right against the thing demanded in the moveable: that is, against 84. shall be the answer in the fixed, viz. 56. and so many pounds will rent yearely the said 84. Acres.

Secondly, note further, that those three numbers as 45. Acres, 30. li. and 84. Acres, are distinguished by numerall attri∣butes, as first, second, and third. Hence of some it is called the rule of three, and the answer to 84. Acres is called the fourth number, which is ever of the same denomination that the second number is of: and the fourth number sought for hath alwaie such proportion to the third number, as the second is to the first: Vel contra.

From which by a more generall name, it is called The Rule of proportion, for that it proportionateth things unto any proportion assigned; so is the said 56. a proportionall num∣ber to 84. as 30. is unto 45. for 56. is two third parts of 84. and so is 30. two third parts of 45.

Therefore these foure numbers, 45. 30. 84. 56. are propor∣tionall numbers one unto another:* 1.1 And here note generally in direct proportion, if the third number be greater than the first number, the fourth number shall bee greater than the second number.

Contrariwise, if the third number be lesse than the first num∣ber, the fourth number is lesse than the second number.

But in Reciprocall proportion this fourth number is inverted; so if the third number be greater than the first, the fourth number is lesse than the second.

[Example 1] So if 45. men in 30. daies, will doe a service, in how many daies shall 270. men doe it.

Here the denominations alike are 45. Men and 270. Men, the answer to 45. men is 30. daies, the answer to 270. men is required.

* 1.2If you move 45. in the moveable to 30. in the fixed, right against 270. in the moveable is 180. daies in the fixed: which answer is absurd, seeing there is more men allotted to doe the worke, there must most necessarily be lesse time.

Therefore in all Questions of Reciprocall proportion, let the demand bee sought out upon the moveable, viz. 270. and brought to the first numbers answer in the fixed, viz. 30. so right against the first number in the moveable, viz. 45. is the answer in the fixed, viz. 5. and in so many daies will 270. men doe that service, if 45. men doe it in 30. daies.

[Quest. 2] Againe, if 3840. souldiers are victualed for 10. moneths, how ma∣ny men may it serve that the said provision may last 12. moneths.

* 1.3In this and all others (as before) bring the third number 12. in the moveable, to the other numbers answer in the fixed, viz. 3840. so against the first number 10. moneths in the moveable, is 3200 men in the fixed, and so many men will the same provision serve for 12. moneths.

From which direction, those ensuing questions, and the like, may be resolved.

[Ques. 3] If I lend 140. li. for 7. moneths, if I should borrow of him 200. li. how long might I keepe it; facit 4. moneths and 9. 10.

[Quest. 4] According to the Statute, if Wheat beat 50. s. the quarter, the penny loafe should weigh 6. ounces and a halfe, what shall it now weigh, if in case wheat be at 3. li. 12. s. the Quarter; the numbers changed into decimals will be thus, if 2. li. 5. 10. give 6. ounces, & 5. 10. what shall 3. li 6. 10. give: facit 4. ounces and 5. 10.

[Quest. 5] A Gallery is found to containe in the walls 380. yards, how many yards of Tapestry shall hang that Gallery of 7. quarters broad: facit 217 yards.

[Quest. 6] 25. Ounces of 7. yards to an ounce will serve to lace a vesture, how many ounces of 5. yards to an ounce will doe the same, &c. facit 35. ounces.

How to proportion a Fraction that is not Decimall, into a decimall.

So if 8. & 12. 40. were to be used, 12. 40. must be changed into a Decimall, thus: bring 40. in the moveable to 10. in the fixed, so right against 12. in the moveable is 3. in the fixed, so the fraction 12 40. is changed now into 3. 10. so for 8. & 12. 40. you have now 8. and 3. 10. which may be easily found out.

Againe, let 63. 84. bee a fraction which is to be used, this cannot be found out upon the Grammelogia: change it there∣fore into a Decimall.

Bring therefore 84. (the Denominator) to 100. in the fixed, so 63. (the Numerator) in the moveable, gives 75. in the fixed; so 63. 84. is now changed into a Decimall 75. 100. the same in value with 63. 84. and so of any other Fraction that is not decimall.

This for Lineary Proportion.

Of the Golden Rule, or Rule of Proportion, in respect of Lines and Quantities in plaine Figures.

[Pro. 1] IF the demand be of the quantity, As if ••he Diamiter of a Cir∣cle be 7. and the Area 38. and 5. 10. what is the Area of another Circle whose Diameter is 18. Foot.

* 1.4Bring the line knowne to the other line, that is 7. to 18. so right against 38. and 5. 10. in the moveable is 99. in the fixed, which looked out in the moveable, right against it in the fixed is 254. and 5. 10. the Area of that Circle.

In like manner consider of Squares, Triangles, and other plaine Figures.

[Pro. 2] If a peece of Land of 20. Pole square be••worth 30. li. what is a peece of Land of the same goodnesse worth, which is 35. Pole square eve∣ry way.

* 1.5Bring 20 to 35. so right against 30. in the moveable you have 52. and 5. 10. in the fixed; and right against this 52. and 5. 10. in the moveable you have 91. and 8. 10. in the fixed, the worth of that land.

[Pro. 3] If a peece of ground of 50. paces square is sufficient to lodge an Army of 1600 men, how ma••y men shall there be ledged in a peece of ground which is 40. paces square

* 1.6Bring 50. to 40. so right against 1600. in the moveable is 1280. in the fixed,

[Pro. 4] Our English land measure is 16. foot and a halfe to the Pole, the Irish Pole hath 21. foot, how many Engl sh Acres doth 30. Irish Acres make.

* 1.7Bring 16. and 5. 10. to 21. then right against 30. in the moveable is 38. and 2. 10. in the fixed, and right against this 38. and 2. 10. in the moveable is 48. and 6 10. in the fixed, and so many English Acres is contained in 30. Irish A∣cres, &c.

* 1.8Our usuall measures in England to the Pole are 16. foot and a halfe 18. or 20. foot, the proportion of their squares are 68. 81. 100. I have set their measures to those numbers in the Grammelogia.

Now if the quantity be given and his measure, and the quantity be required according to another measure, you may have it with greater expedition: for bring the measure whole quantity is required to the other measure, so against the quantity knowne in the moueable, you have the quantity re∣quired in the fixed.

Of the Golden Rule, or Rule of Proportion in respect of Lines, and the quantity of Solids.

[Pro. 1] SO if in some stately structure the Columes were to bee supported with Cubes of Silver, or other rich Materiall, differing in their quantity, an estimate of their charge might be quickly had; As admit the side of the least Cube were 4. Inches, and could not be made under 12. li. what might a Cube of the same mettall be worth that is but one inch more in the side, viz. 5. inches.

Bring 4. to 5. so right against 12. in the moveable,* 1.9 is 15. in the fixed, and right against this 15. in the moveable is 18. and 75. 100. in the fixed, and right against this 18. and 75. 100. in the moveable is 23. li. and 4. 10. in the fixed, and so much will the second Cube cost: this might bee applied to the weight, worth, or quantitie of other Solids.

[Pro. 2] A Pe••ce of 5. Inches boare or Diameter, requires for her charge 16. pound of Powder, what quantity of Powder will serve another Peece of 4. Inches in the boare.

Bring 5. to 4. so right against 16. in the moveable is 12. and 8. 10. in the fixed,* 1.10 and right against 12. and 8. 10. in the move∣able is 10. and 24. 100. in the fixed; and right against this 10. and 24. 100. in the moveable is 8. and 2. 10. in the fixed: the answer of Powder according to Cubick proportion, but Ca∣noniers doe somewhat qualifie this proprotion.

To finde what Proportion in Quantity there is betweene two or more Solids.

[Pro. 3] There are two Bullets, Globes, or Cylenders, the Diameter of the one is 10. inches, and the other the Diameter is 4. inches, what pro∣portion is there betweene the Solids, or how often doth the greater containe the lesser.

Bring 10. to 4. so right against 100. in the moveable is 40. in the fixed,* 1.11 against this 40. in the moveable is 16. in the fixed; and right against this 16. in the moveable, is 6. and 4. 10. in the fixed; so the proportion betweene the Solids are as 100. to 6. and 4. 10.

But how often the greater doth containe the lesser, the Rule ensuing doth teach.

[Pro. 1] How to divide one number by another.

* 1.12MOve the Divisor to 1. so right against the Dividend in the moveable, is the quotient in the fixed.

* 1.13So if it were demanded, how many daies there is in 216. houres, because a day naturall containes 24. houres, that therefore is the Divisor. Move then 24. to 1. and right against the said 216. in the moveable is 9. in the fixed, and so many daies is 216. houres.

* 1.14Here note that in all Divisions, by how many figures or places the Dividend exceeds the Divisor, so many places or figures shall the Quotient have. But if the figures of the Di∣visor may be taken from as many of the first figures or places towards the left hand of the Dividend, then the Quotient shall have one place more.

[Example 2] So if it were further required, how many daies there were in 360. houres, or any other number: the Instrument not moved from his first setting, they are all given at one instant: for right against the number in the moveable, is the answer in the fixed, so right against 360. in the moveable is 15. in the fixed, and so many daies are there in 360. houres.

This note serves onely to know the number of Figures or places in the Quotient, by which the denomination of the first figure of the Quotient may be had.

[Example 3] So if it were demanded how many yeares there is in 14600. daies, there being 365. daies in the yeare: this therefore is the Divisor. Bring then 365. to 1. so right against 14600. in the move∣able is 4. in the fixed, but by the former note ☞ it must be 40. and so many yeares is there in 14600. daies, the Instru∣ment not moved, right against any number of daies, as 5000. 10000. 20000. &c. in the moveable, is the yeares in the fixed. With the same expedition and facility may you divide by fractionall numbers.

Further uses upon Division.

[Example 4] IN a yeare are 52. weeks or 365. daies. If J would know the weekly expences of any yearely summe of money.

Bring 52. to 1. then right against any summe of money in the moveable, you have the weekly expences in the fixed:* 1.15

But if you move 365. to 1. then right against any summe of money in the moveable, you have the daily expences in the fixed.

So if the expences yearely were 1000. li. or the charge of a certaine Company of S••uldiers:* 1.16 right against it according to the note ☞ of Division is 2. li. 7. 10. the daily charge: the Instrument not removed, you may see at one instant the daily charge of 20000. li. a yeare, 50000. li. or 100000. li. a yeare: for right against the charge or expence in the moveable, is the answer in the fixed.

More uses upon Division.

[Example 5] It is said that Land is bought after the rate of 14. yeares purchase: if 14. be therefore brought to 1. right against any summe of mony in the moveable, you have the Annuall Rent in the fixed answerable to that money. And thus you have lying before you a whole Circularity of Numbers, by which at one instant, doe but speake the summe of money, right against it is his Rent.

But if the Rent were given and the Purchase required, it is the inverse of this, and is proper to Multiplication, and the Rule followeth in the next page.

Other uses upon Division to finde the Scale to divide the Meridian line in a Sea Chart, according to any bredth, & to a Latitude assigned.

[Example 6] Let the bredth of the Chart extend from the Latitude of 30. unto 40. the degrees of the Equator answerable to the difference of those Latitudes, according to M. Wrights proje∣ction, are 12. & 24 100. Bring this 12. & 24. 100. to 1. so right against the bredth of the Card in the moveable, you have the Inches, or parts of Inches in the fixed to make your scale by to divide the Meridionall line.

So if the bredth or the Card were 33. Inches,* 1.17 right against it in the fixed is 2. Inches, 7. 10. the largenesse of a degree of the Equator: if the bredth were 24. & 5. 10. right against it is 2. Inches: if 20. & 8. 10. then the bredth of a degree is 1. & 7. 10. if 14. & 7. 10. then 1. & 2. 10. if 8. & 5. 10. then 7. 10. &c.

To multiply one Number by another, or to finde the Product of two Numbers.

* 1.18MOve 1. to the multiplier, then right against the Multi∣cand in the moveable Circle, you have the Product in in the fixed Circle.

* 1.19Here note that the Product of any Multiplication, is ever as many figures or places, as there are places or figures con∣tained in the Multiplicand and M ltiplier, if the two first Fi∣gures towards the left hand being multiplied together have excrescence (that is, if the Product exceed 9) otherwise the Product shall bee one figure or place lesse than there are fi∣gures or places contained both in the Multiplicand and Mul∣tiplier.

* 1.20So if 38. be multiplied by 2. the Product will be but two places: But if the said 38. be multiplied by 5 the Product will be three places, for that 3. by 2. multiplied doth not cres••ere, but the said 3. by 5. doth beare excrescence, viz. more than 9.

This Note is only to give domination to the first figure of the Product towards the left hand, for if the Product have two figures, then the first figure of that Product towards the left hand is ten or tens; if the Product have three figures, then the first figure of the Product towards the left hand is hundreds, &c.

[Example 2] To Multiply 18. by 5. Bring 1. to 5. then right against 18. in the moveable is 9. in the fixed, which by the former note ☞ of observation is 90. which is the Product of 18. by 5.

But if 35. were to bee multiplied by 4. move 1. to 4. so right against 35. in the moved is 140. by the last note ☞.

[Example 3] To multiply Fractionall, as 40. and 5. 10. by 7. and 3. 10. Bring 1. to 7. and 3. 10. so right against 40. and 5. 10. in the moveable is 295. and 6. 10. in the fixed; the Product required.

So to multiply 8. 10. by 5. 10. Bring 1. to 5. 10. so right against 8. 10. in the moveable is 4. 10. in the fixed.

Vses upon Multiplication.

[Example 3] 12. Monetht make a yeare, bring 1. unto it, so right against any monethly expences in the moveable you have the yearely expences in the fixed according to the note ☞: So if the monethly expen∣ces were 75. li. right against it in the fixed is 9. which by the former note ☞ makes 900. if •• 50. li. for a moneth, right against it in the fixed is 1800. li. the yearely charges or ex∣pences.

Other uses upon Multiplication.

[Example 4] 60. Minutes make an houre, bring 1. to 60. so right against any number of houres in the moveable is the minutes of those houres in the fixed.

Further upon Multiplication.

[Example 5] Admit lands be sold at 14. yeares Purchase, bring 1. to 14. so against any Rent in the moveable you may at one instant see the purchase thereof in the fixed, having regard to the for∣mer note ☞.

How to square a Number.

[Example 6] To square 18. bring 1. to 18. so right against 18. in the moveable is 324. in the fixed, the square of the said 18. In like manner may you square whole numbers and fractions, as to square 13. and 5. 10. facit 182. and 25. 100.

How to Cube a Number.

[Example 7] As to Cube 6. and 2. 10. bring 1. to 6. and 2. 10. so right against 6. and 2. 10. in the moveable is 38. and 4. 10. in the fixed, and right against 38. and 4. 10. in the moveable is 228. in the fixed, the Cube of 6. and 2. 10.

Againe to Cube 6. bring 1. to 6. for right against 6. in the moveable is 36. in the fixed; and right against this 36. in the moveable is 216. in the fixed, the Cube of 6. &c.

To finde Numbers in continuall proportion unto any two Numbers assigned.

* 1.21BRing the first number to the second, then right against the second upon the moveable, is the third number in the fixed, and against this third number in the moveable, is the fourth number in the fixed, &c.

* 1.22So if the numbers to be continued in proportion be 2. to 4. move 2. to 4. so right against 4. in the moveable is 8. in the fixed, and 8. in the moveable gives 16. in the fixed, and those numbers, 2. 4. 8. 16 &c. are said to be in continuall Proportion.

[Example 2] Againe, it I would continue a Proportion, as 2. to 3. move 2. to 3. then 3. in the moveable shall point out 4. & 5. 10. in the fixed, and 4. & 5. 10. in the moveable shall give 6 & 7. 10. i•• in the fixed, and so on (if need were) to finde others: and those numbers are said to bee in continuall propor••ion one unto another.

* 1.23The increase or interest of Mon••y from this ground is easily found, seeing the increase of the Money must bee in conti∣nuall Proportion to the Principall, as 100. li. is to his In∣terest.

[Example 3] As if the Propor••ion were to be continued to 40. li. as 100. to 108.

* 1.24Move 100. to 108. then against 40. in the moveable is 43. li. 2. 10. in the fixed: the first yeares Interest and its Principall, & against this 43. li. 2. 10. in the moveable, is 46. li 8. 10. in the fixed, which is the seconds yeares Principall and Interest: in like manner may you proceed to other yeares.

* 1.25The Instrument being at this stay, the eye may denote out at one instant the Interest of any summe of Money: for right against your number in the moveable, is both Principall and Interest in the fixed.

[Example 4] As if it were 27. li. 14. s. (that is, 27. li. 7. 10.) right against it is 30. li. ferè, and so much doth 27. li. 14. s. come to at the yeares end, and so all other summes of money doe offer themselves at one instant to the eye in their resolutions.

To finde a meane proportion, or many betweene any two Numbers given.

MArk what number of equal parts in the fixed is against each of the given numbers, (which equall parts represent the Logarythmes of these numbers, if the Logaryt••mall Index be put unto them, which is a unity lesse then the places of any given number) and adde these Loga∣rythmes, or equall parts together, then take the halfe of that summe, which sought out in the former Circle of equall parts, right against it in the Circle of numbers is the meane proportionall required, or the halfe difference of these two Logarythmall numbers, or equall parts be∣ing added to the lesser Logarythme will give the same, or sub. &c.

But if many meane proportionalls be required divide the differen∣tiall Logarythme of the two numbers, or number of equall parts, by a unity more then the number of meane proportionalls, which quotient being by succession added to the Logarythme or equal parts belonging to the lesser number, doth shew the severall meane proportionalls re∣quired. So if betweene the Cubicke numbers 27. and 64. two meane proportionalls were required, the third part of the difference of equall parts betweene these numbers is 125. which being added unto 431. the equal parts against 27. makes 556. against which in the Circle of numbers is 36. the first meane proportionall unto which 556. againe adde successively the said 125. which makes 681. against these equall parts in the Circle of numbers is 48. the other meane proportionall.

According to the same manner betweene 243. and 1024. foure meane proportionalls might be found, to wit, 324. 432. 576. and 768. Or pro∣portions may be found to a tearme assigned, betweene two numbers, either by augmentation from a greater, or diminution from a lesser.

Of the Extracting of square, and Cubicke Roots, and others.

THe construction of this depends upon the latter, because the square Roote of any number, is nothing but a meane proportion betweene a unity and the given number, the Cubicke Roote is the first of two meane proportionalls, betweene the unity and the Cube proposed▪ the Biqua∣drat Roote, is the first of three meane proportionalls betweene the unity and the given number, &c. But because the former direction specified in finding of meane proportions, adheres unto the way and nature of Logarythmes, it being more facil by them, then so to apply it Instru∣mentally, therefore we will somewhat compendiate that labour by the Instrument alone, and avoyd the search of the Logarythmes of num∣bers, and their partitions, and as an ease for Radicall extractions.

To finde a meane proportionall with more facility then is formerly delivered

Constructio. Marke what number of equall parts in the Circle of e∣quall parts E, in the fixed is against the lesser of the two given num∣bers in the Circle of numbers in the fixed.

then bring the lesser number in the move∣able to these parts in the Circle of e∣quall parts noted with Q betweene

• ...Q A
• ...A Q

if these two num∣bers have

• like places or exceede one another even pla∣ces.
• exceede one another by odde places.

so the like number of e∣quall parts in the fixed in the Circle Q which is against the greater num∣ber in the Circle of equal parts E, being sought out betweene Q A, in the fixed shall right against it point out the meane pro∣portionoll in the moove∣able.

But here note that 10. in this Rule must be accounted to have but one place, 100. to have but two places, 1000. three places, &c.

For the extraction of square and Cubicke Roots more compendiously, thats done by an inspection of the eye onely, as is specified in the a∣foresaid Epistle to the Reader, at the last clause of the use of the Instru∣ment, without motion: but hereafter more in that nature. And by the way, note that in the extraction of square Rootes, the Roote doth con∣taine in places alwayes the just halfe of the places of the number gi∣ven if it hath even places, but if it have odde places, then the Roote hath as many places as the greater halfe comes to.

Now as every two places in square numbers, affords one place for its Roote: so Cubicke numbers, affords for every third place, or ternarie one for its Roote: but if the number have any places above, the ter∣narie or ternaries of places, then the Roote shall be one place more then the number of ternaries.

☞ Note further, that in seeking of meane proportionalls it may bee doubtfull what denominations to give unto it when it is found on the Instrument, which may be discovered in this manner, finde the places that the two extreame numbers given would make if they were mul∣tiplyed together, which the Rule in Multiplication, Pag. 14. will shew you; then having the number of places for the product, the former Rule which doth allude to the places for the square Roote, will tell you what denomination to give the meane proportion sought for.

To finde a meane proportion betweene two numbers.

[Pro. 1] NOte if the two Numbers have like places,* 1.26 or exceed one another by two places, move the numbers to and fro, untill 1 in the fixed be equally distant betweene them, which the divisions in the pricked Circle A B will helpe you; so right against 1. in the fixed, is the mean proportiō in the moveable.

[Pro. 2] If the two n••mbers exceed one another by one, or three pla∣ces, move the numbers to and fro, untill 1. in the fixed bee equall distance betweene them; so right against B in the moveable is the meane Proportion.

Some uses upon meane Proportionals.

[Pro. 1] To find how much is taken in the 100. li. in Loane of mony. If 40. li. be lent for two yeares,* 1.27 and at the end thereof were recei∣ved 48. li. and 4. 10. what was taken in the 100.

Finde a meane proportion betweene 40. and 48. 4. 10. which will be 44 according to the last rule; so right against 48. 4. 10. in the moveable, is 110. in the fixed, which is the Principall and its Interest; so ten pound is taken per centum. [Pro. 2] Pro. 2. In warlike discipline, the weakest place opposed to danger, is supplied with strongest force.

Now there are two companies allotted for two severall services, the one containing 500. Souldiers, the other 320. Souldiers, there is a third place, neither so strong as the lat∣ter, nor so weake as the former, therefore a meane number of Souldiers is thought convenient for the defence thereof: what number shall it be?

Finde a meane proportion betweene 500. and 320. facit 400. and this is a meane proportionall number betweene 320. and 500. and the number of men required.

[Pro. 3] Pro. 3. To finde the Scale that protracted a Plot or Building by.

Let the Rectangle A C be 8. Acres,* 1.28 and let the Scale be sought for by which it was protracted or plotted: With any Scale measure the side A B. admit of 10. parts in an inch, and suppose it make 33. & 33. 100. parts, and A D. 26. & 66. 100. parts, according to which the Area of the Rectangle now is 5. Acres and 56. 100. parts; finde a meane proportion between this and the forme. 8. Acres, which is 6. & 67. 100. and this stands against 1. in the fixed, which represents 10. his scale, but 8. in the moveable gives 12. in the fixed, and such were the parts in an Inch of the scale sought for.

[Pro. 1] How to extract the Square Root by the Grammelogia.

* 1.29LEt 1. in the fixed stand toward you, and seeke that num∣ber to be extracted in the moveable, if it have 1. 3. 7. or 9. places, &c. bring the number towards the left side of 1. in the fixed: but if the number have 2. 4. 6. or 8. places, &c. bring it twards the right side of the fixed 1. and move your num∣ber to and fro, untill 1. in the moveable bee as farre distant from 1. in the fixed, as your given number is from 1. in the fixed: (the equall parts in the Circle A B will helpe you in this) so the number in the moveable right against the fixed 1. is the Root sought for.

Here note that 1. or 2. figures hath but one figure for his Root, 3. or 4. figures hath 2. figures or places for its Root, 5. or 6. figures hath 3. figures for its Root, &c.

How to extract the Cubicke Root.

* 1.30VPon the moveable there are those letters A. B. C. the distance betweene A. B. is divided into 10. equall parts, and each part subdivided: the distance betweene A. C. B. is also divided into 10. parts, and each part subdivided, their uses may be thus.

* 1.31Let 1. in the fixed stand alwaies betweene A and B in the moveable for the Extraction of Cubicke Roots, and move the moveable to and fro, untill that the given number and 1. in the fixed be of like number of parts distant from A in the moveable.

So if the given Number have

• 1. 4. 7. or 10.
• 2. 5. 8. or 11.
• 3. 6. 9. or 12.

Places, &c. the Cubicke Root is right against

• A
• C
• B

In the fixed.

And here note that a number of 1. 2. or 3. places hath but 1. figure for the Root; a Number which hath 4. 5. or 6. places hath but 2. figures or places for its Root; a number which hath 7. 8. or 9. places hath but 3. figures or places for its Root, &c.

Vses upon the square Root.

Pro. 1. There are two square formes, the one is 12. every way, and the other 16. every way, if of those two were made one, how many should it be every way in the side.

[Pro. 1] BY the first proportion,* 1.32 pag. 16. finde a number in con∣tinuall proportion to 16. as 12. to 16. facit 21. & 3. 10. adde this to 12. facit 33. and 3. 10. Then by 1. Pro. pag. 17. finde a meane proportion betweene that 33. and 3. 10. and 12. facit 20. the side of the Square required.

Otherwise square 〈◊〉〈◊〉 12. and 16. according to the eighth example, pag. 15. facit 144 and 256. the summe of those Squares is 400. and the Root Quadrat of it by Pro. 1. pag. r 8. is 20. as before, those extractions serve to wonderfull uses in finding the Diagonals of Rectangles, the Diameters and Axis of Solids, the Area, Difference or Agregate of Figures, as well plaine as solid.

[illustration]

[Pro. 2] Otherwise we might apply the Pro. thus, A B is the bredth of a ditch 16. foot, B C the heigh of a wall 12. foot, the length of a sca∣ling Ladder to reach from A to C. would bee as before 20.

[Pro. 3] A and C are two Townes, Alies West of the Meridian of C 16. miles, and C lies North of the Parallel of A 12. miles, the distances of the two Townes would bee as before 20. miles, &c.

Pro. 4. How to encampe horse or foot, according to any proportion assigned.

[Pro. 4] 240. men or horse are to be imbatled,* 1.33 that the Flanke to the Front shall be in proportion, as 3 to 5, how many shall be in the Front, and how many in the Flanke.

Bring 3. to 5. so against 24. in the moveable is 400. in the fixed,* 1.34 the square Root of which is the Front viz. 20. divide the said 240. by the Front, 20. the Quotient is 12. the Flanke.

In mentall reservation of a number, to finde that number.

[Pro. 5] Let the number be broken into two parts, and to the pro∣duct of the parts adde the square of the halfe 〈◊〉〈◊〉 difference of the parts, the Root, Quadrate of the Agragate is halfe th•• number conceived, &c.

Further uses upon the Grammelogia in the resolution of Questions, touching Interest, Purchases, valuation of Leases, and such like.

* 1.35NOte that from 1. in the moveable, there is charactered 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. &c. all of equall distances, those serve for the number of yeares as occasion requires.

Pro. 1. To finde what a summe of money comes to, at the end of any number of yeares, accounting 8. li. for 100. p••r Annum.

[Pro. 1] * 1.36So if 20. li. were forborne 12. yeares, how much doth it come to allowing Interest compound at 8. li. for 100. li.

* 1.37Bring 1. in the moveable to 20. in the fixed, so right against 12. yeares in the moveable you have 50. li. 4. 10. in the fixed. And so much will 20. li. amount to being forborne 12. yeares.

* 1.38The Instrument not removed, you may at one instant see the amount of the said 20. li. for any number of yeares of parts of a yeare; for right against the time in the moveable, you have the answer of the money in the fixed.

[Pro. 2] Pro. 2. To finde what a summe of money which is due any number of yeares to come, is worth in ready mony, allowing 8. li. for 100. li.

* 1.39So if the said 20. li. were due 12. yeares hence, what is it worth in Present.

* 1.40This is only the converse of the former: bring therefore 12. yeares in the moveable to 20. in the fixed, so right against 1. in the moveable is 7. li. 94. 100. in the fixed, which is a∣bout 7. li. 18. s. 3. d. and so much is able to buy the said 20. li. to be received 12. yeares hence.

[Pro. 3] Pro. 3. A yearely Rent of a Lease, or a Pension to be sold for any number of yeares, to finde the worth thereof in ready money: Or the Rent for any number of yeares being unpaid, to finde what it amounts unto, a••counting 8. li. for 100. li. per Annum.

* 1.41Let a Lease or Pension of 20. li. per Annum be sold for rea∣dy mony, which is in being 12. yeares, how much is it worth? Bring 8. to 100. then right against 20. li. in the moveable is 250. li. in the fixed;* 1.42 unto this 250. in the fixed bring 12. yeares; so right against 1. in the moveable is 99. li. 3. 10. in the fixed, which taken out of the said 250. there remaines 150. li. 7. 10. the worth required.

If the Rent were behind unpaid 12 yeares.

THen bring 1. to the said 250. so right against 12. yeares in the moveable is 630. li. in the fixed, take the former 250. from this 630. li. it leaves 380. li. and so much doth the said Rent of 20. li. per Annum amount to forborne 12. yeares at 8. li. for 100. li. per Annum.

Pro. 4. A summe of money borrowed, and a Lease ingaged for that money, to finde how long the Lease ought to be kept.

[Pro. 4] Let 300. li. be borrowed upon a Lease in being 20. yeares,* 1.43 of 50. li. a yeare, how long shall the Rent be received, that neither be damaged one by the other, accounting 8. li. for 100. li. per Annum.

Bring 8. to 100. so right against 50. in the moveable is 625. in the fixed:* 1.44 from this 265. li. subtract the mony bor∣rowed, viz 300. li. it leaves 325. li. then bring 1. to this 325. in the fixed; so right against the same 625. in the fixed, is 8. yeares 5 10. in the moveable, and so long time shall the Len∣der of the Money enjoy the Borrowers Lease, after 8. li. for 100. li. per Annum. This may be inverted, knowing the Summe and time to finde the Rent.

[Pro. 5] Pro. 5. A Lease to beginne for yeares to come, and then to continue for any nu••ber of yeares, to finde the worth thereof in present, accounting Interest Compound at 8. li. for ••00. li. per Annum.

Let a Lease of 40. li. per Annum beginne 7. yeares hence,* 1.45 and then to continue 10. yeares after; if it were to bee sold, what is it worth in ready money?

By the third Pro. finde the worth thereof in the present for the 10. yeares, facit,* 1.46 268. li. 4. 10. then by the second Pro. finde what that 268. li. 4. 10. is worth in present if it were to be received 7. yeares hence, facit, 156. li. 6 10. and so much is the said Lease of 40. li. per Annum worth, which is to beginne 7. yeares hence, and then to continue unto 10. yeares.

Thus I might have gone further in those matters, but I intended not to be large in this Tract, onely shewing what weighty, and difficult matters in this kinde by the Grammelogia, or Mathematicall Ring, may bee easily and speedily resolved.

Conclusion.

IF there be composed three Circles of equal thicknesse, A. B. C. so that the inner edge of D and the outward edge of •• be•• answerably graduated with Logarithmall signes, and the out∣ward edge of B and the inner edge of A with Logarithmes; and then on the backside be graduated the Logarithmall Tan∣gents, and againe the Logarithmall signes oppositly to the for∣mer graduations, it shall be fitted for the resolution of Plaine and Sphericall Triangles.

* 1.47So if you move the Signe of 90. Degrees vnto the Tropicall point in the fixed, you have the Declination of any Degree of the Eclipticke onely by an ocular inspection, for right against the Sunnes longitude in the moveable amongst the Signes, is the Sunnes declination in the fixed.

Againe, in the 〈◊〉〈◊〉 of Tangents, if you bring the comple∣ment of any Latitude in the moveable to 45. in the fixed, you may at one instant have the time of Sun rising or Sun setting for any Declination required in that Latitude; for right against the Tangent of the Sans Declination, you have the fine of the Suns ascentionall differen••e: and in plaine Triangles the opera∣tions are performed with like facility.

Hence from the forme, I have called it a Ring, and Gram∣melogia by annoligie of a Lineary speech; which Ring, if it were projected in the Convex unto two yards Diameter, or there a∣bouts, and the line Decupled, it would worke Trigonometrie un∣to seconds, and give propo••tionall number•• unto six places only by an ocular inspection, which would compendiate Astro∣nomicall calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability so health and time.

FINIS. This Instrument is made in Silver, or Brasse for the Pocket, or at any other bignesse, over against Saint Clements Church without Temple Barre, by Elias Allen.

Notes

• * 1.1

  ☞
• * 1.2

  Constructio.
• * 1.3

  Constructio.
• * 1.4

  Constructio.
• * 1.5

  Constructio.
• * 1.6

  Constructio.
• * 1.7

  Constructio.
• * 1.8

  ☞
• * 1.9

  Constructio.
• * 1.10

  Constructio.
• * 1.11

  Constructio.
• * 1.12

  Constructio.
• * 1.13

  Declaratio.
• * 1.14

  ☞
• * 1.15

  Constructio.
• * 1.16

  Declaratio.
• * 1.17

  Declaratio.
• * 1.18

  Constructio.
• * 1.19

  ☞
• * 1.20

  Declaratio.
• * 1.21

  Constructio.
• * 1.22

  Declaratio.
• * 1.23

  ☞
• * 1.24

  Constructio.
• * 1.25

  ☞
• * 1.26

  ☜
• * 1.27

  Declaratio.
• * 1.28

  Declaratio. Constructio.
• * 1.29

  Constructio.
• * 1.30

  ☞
• * 1.31

  Constructio.
• * 1.32

  Constructio.
• * 1.33

  Declaratio.
• * 1.34

  Constructio.
• * 1.35

  ☞
• * 1.36

  Declaratio.
• * 1.37

  Constructio.
• * 1.38

  ☞
• * 1.39

  Declaratio.
• * 1.40

  Constructio.
• * 1.41

  Declaratio.
• * 1.42

  Constructio.
• * 1.43

  Declaratio.
• * 1.44

  Constructio.
• * 1.45

  Declaratio.
• * 1.46

  Constructio.
• * 1.47

  Example.