Question 1.

If 1 lb of Pepper cost 1 s. 10 d. how much will 1 C & 8 lb come unto? facit 11 lb. First 1 C & 8lb is 120 lb. then say 120 s. is 6 li. and 120 times 6 d. is half so much, that is 3 li. and 120 groats is 2 li. so the totall is 11 li. and stands in order thus:

Question 2.

If 1 yard of black Cloth cost 1 lb 19 s. what shall 30 yards of this Cloth cost at the same rate? Here is 30 lb. 30 Angels, 30 Crowns, 30 half Crowns, 30 Shillings, and 30 Testers; which collected to­gether is 58 lb. 10 s. as by this following example:

Question 3.

This Rule of Practice is lesse vul­gar,

yet more artificiall, and of greater use than is the former, and solves divers questions, which the other cannot do without some men­tall reservation; for here in this Question 30 lb of Commodities cost 45 L sterling, and it is required what half a C or 56 pound weight will cost? According unto the Rule of Three direct, the product or 45 & 56 divided by 30 will produce for the fourth number 84; now by this Rule & the 13 Axiome this Question may be solved with lesse Multipliers, and without a Divisor, that being first reduc'd unto an unite, as thus, divide 30 lb by 5, See Lib: 1. Sect: 1. Parag: 5. Exam: 10. & Lib: 2. Parag: 7. Axiome 13. And so likewise 45 L. being one of the [Page 214] Multipliers, the reduction will be 6, 9, 56, then di­vide 6 & 56 by 2, the Quotient will be 3, 9, 28, then divide 3 & 9 by 3, and so you will produce these 3 numbers, viz: 1, 3, 28, in the same propor­tion that the Question was stated in, and 28 lb mul­tiplied by 3 L. produceth 84 L. the solution of the Question, as was required.

Question 4.

In all Questions of this kinde in the Rule of Three, having reduced the Divisor unto an unite, if the se­cond number (on which the Proposition stated does depend) shall consist of severall denominations, di­vide them into what parts you please, as in the first Quest: of this Parag: but best into such parts as may successively depend one upon another, either in number, weight, or measure, as in these following Examples: A man had made a piece of Cloath that stood him in 1 li. 12 s. 2 d. farthing a yard, and it is required to know what 30 yards would come unto at the same rate? In the first place take 1 li. then 10 s. as the half of that, then 2 s. as the ⅛ of the last: next 2 d. the 1/12 part of 2 s. and lastly, ⅛ of 2 d. for the Farthing, the totall will be 48 li. 5 s. 7 d ½, or divide the price into these parts, according unto this example,

Question 5. If 10 yards of Holland did cost 1 L. 3 S. 6 ½ D. What will be the price of 1 yard?

In the Rule of [...] Practise, where an Unite is the Multiplier, the o­ther number is to be divided onely by the first in the Rule direct, which is commonly an unite with cy­phers annext unto it, so there is nothing more to be done in such cases than to cut off so many figures upon the right hand of the first denomination, as there be cyphers in the Divisor, and if in case the Dividend does prove the lesser number, reduce it into the next denomination lesse, and having added in their parts; (if there be any) cut off so many places as before, and reduce that remainder unto the next, and so proceed: but to avoid mistakes, and to have your worke before you, observe the so­lution to this question and forme of the Table, where first I state the question again, as 10 Y. cost 1 L. 3 S. 6 D ½, what costs 1 Y? The third being an unite, and the first denomination of the middle num­ber is 1 L. in which 10 the Divisor cannot be con­tained; reduce it to Shillings, and then you will finde 23 S. place 3 S. on the right hand of the line drawn in the middle of the Table: this 3 S. is 36 D. & 6 pence added to it, the summe will be 42 D. place 2 D. on the right hand of the Table, and 4 D. [Page 216] on the left: the last remainder is 2 D. which is 8 Farthings, to that adde ½ D. the summe is 10, place the cyper on the right hand, the question solved; and 1 Y. cost 2 S. 4 D. ¼ just, as in the Table. See Lib. 1. Sect: 1. Parag: 5. Example 10.

Question 6. If 10 yards of Linnen cloath cost 2 L. 14 S. 9 D ½: what will the price be of 6 yards?

In this question

10 yards is the Divisor, and up­on 6 yards is the querie made, w^ch multiply into all the severall deno­minations of the middle numbers, which is the price propounded, the product will be 16 L 8 S. 9 D. and being 10 is the Divisor, place 6 upon the right hand of the line, and 1 L. on the left, which is the Quoti­ent; then reduce the 6 L. (which remains) into Shillings, and adde 8 S. unto it, the summe is 128, which subscribe with 8 S. placed upon the right hand of the line, those converted into pence with the 9 D. added to it, makes the summe of 105 pence, which set down, placing 5 D. upon the right hand of the line, which reduced to the next denomination lesse will be 20 Q. which divided by 10, as before, by placing the cypher upon the right side of the line, the Quotient is 2, and so if 10 yards cost 2 L. 14 S. 9 D ½. then 6 yards of the same cloath will cost 1 L. 12 S. 10 D ½, the question solved as was required.

If an unite shall be one of the Multipliers, and the Divisor a greater decimall, then so many figures or cyphers must have been placed upon the right hand of the line, as the Divisor shall have cyphers, but if they are not annext unto an unite, but as 20, 30, 400, &c. or all significant figures; a division must be made by those past and the succeeding ex­amples, this Rule of Practice will be sufficiently il­lustrated to the ingenious.

Question 7. If any certain number of Men shall perform a piece of worke within a known time; in what time would a greater or lesser number of men have performed the same?

The Question here stated is

how 28 Men performed a piece of worke in 12 Dayes, and it is required in what time 8 Men would have finished the same worke; it is evident that the fewer Men must have the longer time, and consequently performed by the Rule reverse, 8 being made Divisor, which must be reduced unto an unite (if performed by the Rule of Practice) and may be thus reduced, 8 may be divided by 4; and also 28 (one of the Multipliers.) So in the second row you will finde 7, 12, 2, then divide 2 & 12 by 2, and then you will finde in the third row these 3 numbers, viz: 7, 6, 1, the product of the two Multipliers will be 42 Dayes, the Que­stion solved as in the fourth row; if 8 Men should [Page 218] have performed the same worke in 12 Dayes, and it had been required in what time 28 Men would have performed the same, the Question must have been solved by the Rule of Three reverst, because the greater number of Men would have required a shorter time, viz: 8, 12, 28; or divide 12 & 28 by 4, then it is as 8, 3, 7, then the first and last divide by 7 and it will be 1 1/7 or 8/7 3 & 1, that is 3 3/7 Dayes, and at 12 houres to the day, the time is 3 Dayes, 5 Houres, 8′, 34″ &c. But all Questions of this kinde reduced into fractions are as readily perfor­med by the Rule of Three, as by this of Practice; yet some I will here insert when an unite is Mul­tiplier. See the 5 & 6 Examples.

Question. 8. If part of any number, weight, or measure, did cost a part of any summe or integer of money, what shall any quantity of that commodity cost?

In the first place here is ¾

of a yard of cloth, which cost ⅝ L. and it is demanded, what 78 yards of the same cloath will cost, at the rate propoun­ded? according to the former Rules in Fractions, reduce them unto one common De­nominator, as in the second row, 24/ [...]2 & 20/32 to 78. See Lib: 1. Sect: 2. Parag: 1. Parad: 8. now according to the former Rules, the Denominators being alike may be omitted; then in the third row, the proportion is as 24 unto 20, so [Page 219] will 78 be to a fourth proportionall number: but to return, these 3 numbers may be reduced by divi­ding 24 the Divisor, and either of the Multipliers by the same number, according to the 13 Axiome, Parag: 7. and here 24 & 20, divided by 4, the fourth row will be 6, 5, 78. then divide 6 & 78 by 6, the fift row will be 1, 5, 13, which 5 L. multiplied by 13 yards produceth 65 L. the solution of this question, as in the 6 row.

Question 9. When the Divisor is a Fraction, or compound number in the Rule of Three direct or reverse; it may be reduced to the Rule of Practice by this following Example.

The state of this Questi­on

is how that cloath 6/4, or 6 quarters broad: ⅞ yard made a Childes coate, and it is required to know how much Stuffe will make the same Childe a coate when the Stuffe is but ½ yard wide? which being narrower than the cloath, it is evident a greater quantity must be required, and consequently performed by the Rule reverse: the Question will stand as in the Example 6/4 or 3/2 re­quires ⅞ yar. what at ½ yard broad?

Question 1. There was a Knight had 5 Sonnes, and 3 of these in 4 moneths had spent him equally 19 L. that is in all 57 L. and it is required what his 5 Sonnes would have spent at that rate in 12 moneths?

In this double Rule [...] of Three there are 5 termes or numbers gi­ven, whereof one is simply of it selfe, viz: the money spent, which denomination is the proportionall number required, and in the Rule of Three direct, as by the Example; for if the 3 Sonnes did spend 57 L. then his 5 Sonnes would have spent a greater summe, as 95 L. and if the space of 4 moneths (as by the second Rule) requi­red 95 L. then 12 moneths (in a direct proportion) will require 285 L. as in the Example, the Question solved; being the true expenses of the 5 Sonnes in 12 moneths, at the same rate, as the 3 Sonnes in 4 moneths time did spend 57 L. and as in this, so in all other Questions of this kinde, according to my former directions, it is not materiall whether of these two Rules are first, so long as the single terme be made the mean proportionall number in the first Rule: as 4 M. to 57 L. so 12 M. to 171 L. then in the second Rule it will be; as 3 is to 171 L. so 5 unto 285 L. as before.

Question 2. If 12 Pecks of Provender did serve 4 Horses for 3 dayes, how long will 24 Pecks serve 3 Horses at the same dayly allowance.

In such Propositions [...] two Rules of Three are contained, the one di­rect, the other reverst; as in this Example, the first Rule is in a direct proportion, viz: as 12 pecks to 3 dayes; so will 24 pecks, be in pro­portion unto 6 dayes, a mean proportionall num­ber for the second Rule, in which 4 Horses must be the first terme whereon the Question depends, be­ing of the same denomination with the third terme, upon which the demand is made, and performed by the Rule of Three reverst, it is evident, because the fewer number of horses must require a longer time; so it will be as 4 horses to 6 dayes, so will 3 horses be in a reverst proportion unto 8 dayes; and so long time will 24 pecks of provender serve 3 hor­ses; that is a peck allowance every day for a horse, the thing required.

If this double Rule of Three (performed at two operations) shall fall in fractions or mixt numbers, either in the Rule direct or reverse, by these last Examples it may be solved, with help of the 8 Parag: to which I referre you; onely observing in all double Rules of Three compounded (as in this last [Page 223] Example) which of the two Rules is reverse, and which direct; this done, (as reason will dictate) it is not materiall whether of them be solved first, as 4 H. to 3 D. so 3 H. in a reverst proportion unto 4 D. Secondly, as 12 P. to 4 D. so 24 P. in a direct proportion to 8 D. as it was before: so in either Rule, the termes on which the demand is made, must alwayes be the third number, as these were, viz: 3 Horses & 24 Pecks: so I will say no more of this double Rule performed at two operations, which by one Rule will be solved with more facility in lesse time by these following Examples.

Question 3. If a mans 3 Sonnes spent 57 L. in 4 moneths, what would his 5 sonnes expences have been at that rate in the space of 12 moneths?

The first question I have

here stated again, to satisfie the Reader, in the operation of this double Rule at one worke, by a single Rule of Three at most, if not reduc'd unto a Rule of Practice, as in this Example; where in the first Rule stands 3 S. 57 L. & 5 S. In the second Rule is placed for the two extremes the time, viz: 4 M. & 12 M. the demand is of the 5 S. and the 12 M. The other 3 numbers were proposed, viz: 3 S. 57 L. 4 M. if 57 L. were multiplied by 5 and divided by 3, the Quotient would be 95 for the fourth number, and a meane proportionall in the [Page 224] second Rule, and that multiplied by 12 and divided by 4 will solve the Question: then for brevity, since the number or summe here required is contained in 57 L. twice multiplied, and as often divided, I say if the two Multipliers, multiplied into one Multiplier, and the two Dividers into one Divider, the propor­tions must be the same; therefore by the 14 Axiome this Example with two Rules will be reduc'd to one, the product of the Multipliers is 60, and the Divi­sors 12, so the proportion is as 12 to 57 L. so 60 to 285 L. or reduced by the Rule of Practice, as 1 to 57 L. so 12 will be in proportion unto 285 L. as before, the Question solved.

Question 4. If 4 Horses did eat 12 Pecks of corne in 3 dayes, in what time will 3 horses eat 24 pecks of the same corne and dayly allowance.

This differs no­thing [...] from the se­cond Question, but in the operation it being reduced unto a single Rule of 3, & in this manner: Here you may ob­serve two Rules, whereof one is di­rect, the other reverst; all which rightly compre­hended, the reason is the same as in the 3 Quest: of of this Parag: the forme onely differing: for here note, the first Rule is direct, the second reverst, and [Page 225] not materiall whether of the extremes in either Rule be multiplied crosse-wise, viz: 12 by 3, and 24 by 4, their products are 36 & 96, and 3 the medium: so the proportion is now direct, as 36 unto 3, so will 96 be to 8 a fourth proportionall, and the number of dayes that 6 bushels or 24 pecks will serve 3 horses, with the same allowance that 3 bushels did serve 4 horses. Here note that 12, and 3 in the second Rule were both Divisors; and like­wise 24 & 4 were in both Rules Multipliers. And farther observe that if the first Rule had been re­verst, and the second direct, their products set un­derneath their Multiplicands, it is alwayes reduced to a Rule of 3, and of the same species with the first Rule, and the products placed under the Multipliers, is ever the contrary, and of the same species with the second Rule.

Question 1. Two men, viz: A and B, joyned their stocks together: A had 18 L. and B had 12 L. with which they bought wares, and quickly sold them againe for 2 L. 10 s. profit: how much did either of them gaine?

According to

the Rule before specified 30 L. the adventure stands in the first place of the Rule of Proportion: the middle number or mean is the whole gains, and in the third place stands the adventures, as the share of A 18 L. and B 12 L. these parts must ever be multiplied by the gain or losse, and divided by the whole adven­ture: as 18 L. encreased by 50 s. produceth 900 which divided by 30, gives 30 s. for the gains of A. So 12 L. multiplied by 50 s. and divided by 30, sheweth 20 s. for the gains which B hath made clear profit: to prove this Rule worke them all back­wards, and you will finde the first adventure by eve­ry one of them; or for more brevity observe the particular shares, with their gains, the summe of those particulars, must be respectively equall to the totall adventure with the gain or losse.

Question 2. Five Merchants put their stocks together, as A, B, C, D, & E: whereof A did adventure 175 L. B. 140 L. C 70 L. D 35 L. E 30 L. this stock re­turned them 744 L. all charges defrayed: How much was each mans gain or losse?

In the first place

collect all the particular stocks, or adventures toge­ther, which is here 450 L. and being it is lesse than the stocke returned, it is evident they were all gainers by the adventure, and in proportion according to each stock adven­tured, as in the last example; but here note, that all the three termes, are of one denomination, viz: pounds sterling, yet differing in their qualities, the first number being the total stock, and the third number, each particular stock, the mean proportion, viz: 744 is the totall return of the adventure, to which a fourth number is required, to every parti­cular stock; which here will be found with the pro­fit together, because the middle number is of the same quality, viz: the whole stock as 450 L. and the profit of the adventure, as 294 L. so in these two examples there is no materiall difference, for in either way the encrease of their stocks will be discovered by Addition, or Subtraction: for ha­ving multiplied each particular stock by 744, and [Page 229] the products divided by 450, the quotient will be each mans gain, and his stock adventured too; as the share of A comes to 289 L 6 s 8 d. so the cleer profit he made of 175 was 114 1/ [...] L. and so for all the particular adventures, as against each letter in the example will appear.

Question 3. Three men unhappily imployed their stocks together, as A, B, & C: whereof A deposited 20 L. B 30 L. C 60 L. all which they layd out; and afterwards sold those goods for 84 L. 6 s. 8 d. how much was each mans gain or losse?

As in the for­mer

Example, set all shares downe in the third place, the summe of thē in the first place, the money returned in this adven­ture must be the middle number as before, which is here 84 ⅓ L. which is lesse than the adventure or totall stock, from whence it is evident that the mo­ney was imployed ill, and returned with losse, and consequently the fourth termes found must be lesse, than was each particular; share which to finde, dif­fers nothing from the former, onely the middle terme being a compound number, must be reduced into one denomination, as into Groats, which will be 5060 Gr. or made an improper fraction, as 2 [...]/ [...] L. and work it according to the Rules of Broken num­bers, or in any other parts, that are readiest for use, [Page 230] as in this I will omit the fractions denominator, and so the summe, or stock returned, is 253 Nobles, which multiplied by each particular share, and divi­ded by the totall adventure, will produce these parts as in the Example, viz: A must have 46 N. B 69 N. C 138 N. so A had 15 L. 6 s. 8 d. and lost 4 ⅔ lb. B lost 7 L. and C 14 L. the Adventure being double to B 30 lb.

Question 4. Three men held a pasture in common (as A, B, & C) for which they paid 44 L. per annum: A kept 10 steares upon the ground 20 weeks, B fed 15 steares 16 weeks, and C kept 20 there 8 weeks: what rent must each man pay?

In all questions of this

kinde, observe double termes, as in respect of each particular stock & time, upon which all questions of this kinde depend; and to effect this, or the like: multiply each mans termes together, as here the stock and the time, viz: for A 10 steares by 20 weeks, the product is 200; for B 240; and the product for the stock of C and his time, as 20 by 8, will be 160: the summe of these 600 for the first terme, the rent 44 L. is the middle terme, and the three last num­bers are A 200, B 240, C 160, these multipliers, with the divider 600 you may reduce by 40 unto 15 the first terme: the other three will be 5, 6, 4; the meane proportionall might have beene reduced [Page 231] with the Divisor, but not so conveniently, the three multipliers being made simple figures, the readier much for use, yet as true without reduction, as by the 13 Axiome, Parag: 7. the parts or shares here to be paid for rent are these, viz: A 14 L. 13 s. 4 d. for B 17 L. 12 s. and for C 11 L. 14 s. 8 d. all which summes added together doe make 44 L. the whole rent, which shews the work is true: as by the Example does appear.

Question 5. Four men joyned their stocks together, as A, B, C, & D, whereof A ventured 40 L. B 160 L. C 100 L. D 280 L. by misfortune they were all losers: upon which they fell at difference, at severall times, and broke off from this society, when D had lost 20 L. C 10 L. & B 30 L. A continued the trade 12 moneths, and lost 8 L. how long were all their stocks continued at that rate or proportion?

In all questions of

this nature, the dou­ble proportion must be made one, as here in this it is A, whose stock was 40 L. the time 12 moneths, the product 480; his losse was 8 L. for the first terme, so the proportion will be, as 8 L. losse is unto the product of the time and principle, that is here 480; so shall each particular losse be proportionable unto the product of his time and principle, and being it con­tains them both, and the adventure known, divide that fourth proportionall found by his principle or [Page 232] stock, and the quotient will discover the true time; as in this last Example, 480 multiplied by 30 L. (the losse which B sustained) the product will be 14400, which divided by 8 L. the first terme, the quotient will be 1800 the true product of time and principle; therefore 1800 divided by 160 L. the adventure of B, the second quotient will be 11 ¼ moneths, that is, 7 dayes in all 45 weeks; and so long time did B con­tinue his stock in the same company; in this man­ner, C that lost 10 L. will be discovered 6 moneths, and D that lost 20 L. kept in this society but 4 mon: and 8 dayes, as in the preceedent Example is evi­dent, where 68 L. was lost in all.

Any question of this kinde, must be tried by a contrary way; as with each mans adventure, his time, and the losse of one mans stock, to finde the others: and so likewise in any other question where­in gain is made: and no more questions will I shew here in this Paragraph, lest that I should lose more time, than the Reader shall gain benefit.

Question 1. A man mixt for provender 3 quarters of beanes at 3 s. the bushell: 3 ½ quarters of pease at 2 s. 6 d. the bushell; with 5 ½ quarters of oates, at 1 s. 6 d. the bushell: what will a bushell of this mistling be worth, when all these are mixt together.

Reduce all

the measures into one de­nomination that is least, & against every quantity set downe the price, or va­lue of them, as in the margent; then take the totall summe of both, as in this Example, the quantity mixt is 96 bushels, and the totall of their severall prices, is 10 L. 8 s. or 208 shillings: these place, as at A, according to the Rule, as the quantity ninety six bushels shall be to the price, 208 shil­lings; so will 1 bushell be unto the price thereof, viz: 2 ⅙ s. that is 2 s. 2 d. the bushell, the common price required: to trie any such question, whether it be true or no, the quantity multiplied by the com­mon price (if true) will be equall unto the summe of the particular prices; as here 96 times 2 s. 2 d. will be 10 L. 8 s. equall unto all the particular prices.

Question 2. A Vintner put into a Wine-casck 15 gallons of Canary-sack which cost 4 L. 10 s. 30 gallons of Mallage 6 L. 5 s. & 35 gallons of Sherry sack 7 L. with 40 gallons of Greek wine 8 L. a pipe thus filled, the price of one gallon is demanded?

The quanti­ties,

and price of the severall sorts of wine, being set down as in the Table here, take the totalls of them as 120 gallons which cost 25 L. 15 s. or 515 shill: these place in the common Rule of 3 as at A. viz: if 120 gall: cost 515 shill: what shall 1 gallon cost? the fourth proportionall num­ber, or common price will be discovered 4 s. 3 ½ d. and will be proved, as was the last example, for 4 s. 3 ½ d. multiplied by 120, (the number of Gallons) and the product will be reduced unto 515 s. or 25 L. 15 s. as before: to finde what the particular sorts cost a Gallon, or what the common price of this mixture is a quart, or in any other measure: were a question onely fit for young beginners: so I will say no more in Questions that onely concerns a common Medium, but proceed.

Question 3. A Druggist had two sorts of Tobacco; the one was Spanish, at 10 s. the lb; the other Virginia, at 3 s. the pound; of these two sorts, he was to mix 112 lb weight, and so, as that it might be afforded for 8 s. the lb: how much must be taken of either sort?

This second

rule differs much frō the former, which, requires onely a common price, from the totall of severall quantities mixt together; whereas this is confined unto a price and quantity in generall, composed of particulars, from whence the mixture is to be made, and the parts taken, proportionally according to each price, quantity or quality; as in this Example, in the Table at A, where 8 s. is the price determined on; the best Tobacco was 10 s. the pound, the worst 3 s. the pound; which two prices I set one above [Page 237] another, and couple them together with an arch of a circle; the price set must be alwaies lesser than the greatest price, and greater than the least of the particulars, as here 8 is lesser than 10, and greater than 3: this done the difference between the price set, and each particular must be found, and counter-changed with the number, to which it is coupled, as in the first Table, the difference betwixt 8 & 10 is 2, which is placed against 3. the difference between 8 & 3 is 5, which stands against 10. the summe of the differences is 7, which according to the Rule a­bove, and the Table, at B, stands in the first place: 112 pound, the quantity to be mixt the second number; and each particular correspondent diffe­rence the third number: so it is now in the Rule of proportion, as 7 the totall difference, to 112 pound the totall quantity; so will 5 (the particular respective difference for the best price) be to 80 pound, the quantity to be taken of that sort. Again, as 7 to 112 pound, so 2 unto 32 pound the quan­tity of the worst sort; and thus repeating the Rule of 3, so often as there be particular differences, you will produce particular quantities to them, whose totall must be equall to the summe, or quantity pro­pounded to be mixt, if otherwise, the operation is false.

Yet least in the mixture of these simples, you should remaine ignorant of the reasons; this questi­on, and all of this kind hath relation unto the 15 and 16 Axiome, parag. [...]7. wherein the totall difference of prices, hath proportion to the totall summe or quantity to be mixt, as the particular difference or parts, have to their respective quantities, but here [Page 238] another Querie will be made, wherefore these dif­ferences between the price set, and each particular price is counterchang'd with a greater and a lesser; by the difference of prices, will be discovered the severall quantities in proportion to those differen­ces; if the difference were equall, the quantities to be taken of all those sorts would prove alike: if the rate or proportion propounded, inclines unto the greater price given, it will have the lesser difference, by how much the neerer it comes to the greatest price, and yet the greater quantity of that sort must be in the mixture; and so consequently the least price (in this case) will have the greatest difference, in which the least quantity is required; and the con­trary will be found by the rule of 3 direct; as by the last example, and the first table at A, where the price propounded is 8 s. the two prices given are 10 and 3, and the differences are 2 and 5: and being 8 is neerer 10 then 3, the greatest quantity of the best sort will be required; and the lesse of the wor­ser sort; therefore as 7 to 112, so 5 unto 80, and as 7 to 112, so 2 unto 32.

In the other Table at C admit a composition of the same commodity were to be made at 6 s, with any quantity assign'd, 6 the price propounded is neerer 3 then 10, therefore 3 must have the greater difference, whereby to produce the greater quan­tity, and the best sort at 10 s. the least difference as 3. the worst sort 4. and so the proportion would have been, as 7 to 112, so 3 unto 48 lb. or as 7 to 112 lb so 4 unto 64 lb both of them making the just quantity of 112 lb and for a farther triall, the price stated, if multiplied into the quantity for to [Page 239] be mixt, the product will be equall, unto the pro­ducts of the severall prices, and their respective quantities: as 6 multiplied by 112 lb. produceth 672; so 48 by 10 is 480. and 64 by 3 maketh 192, which added unto 480 the summe is 672, or 33 L. 12 s. as before.

Question 4. A Grocer had 3 sorts of Sugar, one was worth 6 d. the pound, another sort 10 d. a pound, and the best 15 d. a pound, out of these sorts, he made two parcells: either of them 56 pound weight, the one thus mixt to be afforded at 12 d. the pound: the other sort at 10 d. the pound. How many pound must be taken to be mixt of either sort?

First to prepare the worke, set downe the price propounded, as 12 d. and likewise the prices given, viz: 6 d. 10 d. & 15 d. against these I prefix let­ters, as A, B, & C, and according to the last Que­stion couple a greater with a lesse, whereof A is onely greater, which must then be coupled with B 10, and C 6, and the differences transcribed and counterchanged; so often as there be numbers [Page 240] coupled to any one; as here the difference of prices betwixt C and 12, and also B and 12, are 6 & 2; which differences must be counterchanged with A. and the difference betwixt A & 12, which is 3; must be twice transcribed, as placed against B, & C. the total of differences is 14. thus the first que­stion is prepared; and as for the second propositi­on, there are two numbers the one greater than the price propounded, the other lesse, viz: A 15. and C 6. with either of which you may couple the other two, and is indifferent in respect of the proposition, although the quantities will not be the same, in this I have again coupled them both with A; to a­voyd fractions, wherein I finde the difference be­twixt 10, the price propounded, and B & C to be 4 & 0. which must be placed against A, and the dif­ference betwixt A & 10 the price propounded, for to be 5, which transcribed against B & C. the summe of these differences 14, as before.

The differen­ces

being found, and the rule pre­pared, as before; the proportion is as 14 (the totall of the differen­ces) to 56 pound (the summe or quantity for to be mixt) so will 3 the respective difference for C, be unto 12 pound, for its quantity, at 6 d. the pound; and likewise [Page 241] 12 pound of that sort for B, at 10 d. the pound. Then again, as 14 to 56 pound, so 8 to 32 pound: and so much must be had of that sort at A, of 15 d. the pound: all which 3 quantities makes 56 pound, as in the Table, which argues it is true, or may be thus proved: the common price propounded was 12 d. a pound, so 56 pound the quantity mixt comes unto 2 L. 16 s. now for the particulars, A must have 32 pound, at 15 d. the pound, that is, 2 L. 0 s. then B 12 pound, at 10 d. the pound, comes to 10 s. Thirdly, C must have also 12 pound, which at 6 d. the pound, comes to 6 s. the totall summe 2 L. 16 s. as before.

In the second composition or mixture, it will be as 14 to 56 pound, so 4 unto 16 pound, for the part to be taken of A. Again, as 14 to 56 pound, so 5 to 20 pound: for B, and so much for the composition must be taken of that sort at 10 d. the pound. then as 14 to 56 pound, so 5 unto 20 pound, the quan­tity of C. for being the Multiplicators are alike, C must have also 20 pound, at 6 d. the pound, the summe of these is 56 pound, in money 2 L. 6 s. 8 d. and so proves the particulars; for 16 pound of A, at 15 d. the pound, comes unto 1 L. 0 s. then 20 pound of B, at 10 d. the pound, is 16 s. 8 d. Lastly, 20 pound of C at 6 d. the pound comes unto 10 s. the totall of the compound 56 pound, in money (according to their severall rates and quantities) 2 L. 6 s. 8 d. and so much the 56 pound did come unto, at the set price, viz: 10 d. the pound: had A & B been coupled with C, the proportions had been changed in respect of their quantities, as by the fol­lowing Example will be manifested.

Question 5. A Grocer hath 4 sorts of currans; one at 4 d. another 6 d. a third sort 9 d. the best 11 d. the pound: the worst sort slighted as too meane, the best thought too deare, so two sorts would not sell: upon which the Grocer compounded two parcells out of the 4 sorts, each containing 120 pound: and so mixt, as that he might afford them at 8 d. the pound. How many pound had the man of every sort, in either parcell?

Here I make two Tables again: in either of them set down the price propounded in this 8 d. next in­sert the severall prices, viz: 4 d. 6 d. 9 d. & 11 d and so likewise in the second Tables: which prices you may note with letters against them, as A, B, C, & D. next couple a greater than the price set with a lesse, as in the first Table, 4 with 11, and 6 with 9; In the second Table 4 with 9, and 6 with 11: this done, take the difference between the price pro­pounded, and the severall prices, and transcribe. those differences, with counterchanging their places as in the former examples, and here are found to be in the first Table, A 3, B 1, C 2, D 4. In the second [Page 243] Table A 1, B 3, C 4, D 2: the totall in either is 10. the quantity to be mixt 120 lb then say, As 10 the totall difference is unto 120 lb. (the quantity to be compounded) so shall 3 the respective diffe­rence be to 36 lb, the quantity of the worst sort for the mixture, that is of A, at 4 d. the lb. Again, as 10 to 1 [...]0 lb. so 1 unto 12 lb. the quantity of that sort for B at 6 d. the lb. As 10 unto 120 lb. so 2 unto 24 lb. for the quantity of that sort at C. As 10 unto 120 lb. so will 4 be to 48 lb. for the quan­tity of that sort at D, which is at 11 d. the lb. all which quantities inscribe against each peculiar price, or respective letter, as here in the Table, whose summe must be equall unto the quantity for to be compounded; and the whole quantity mixt, at the price set, to be equall unto the summe of each par­ticular price and quantity, as the totall of the parti­culars in this examp [...]e is 120 lb. which at 8 d. the lb comes unto 4 L. now for the particulars, A 36 groats or 12 s. B 6 s. C 18 s. and D 2 L. 4 s. the totall 4 L. as before.

Now for the second composition, the proportion will be as 10 unto 120 lb. so 1 for A 12 lb. and so proceed as in the first mixture repeating the Rule of 3 so often as there be differences of severall prices, & so these quantities will be found B 36 lb. C 48 lb. D 24 lb. the totall 120 lb. and these particulars, at their own rates will be 4 L. as before.

Question 6. A composition of severall simples to be made in weight 1 ½ C grosse; and to be afforded at the price of 5 s. the lb the particulars at these rates: viz: A 1 s. lb B 3 s. lb C 5 s. lb D 8 s. lb E 11 s. lb & F 14 s. lb How many pound must be taken of each sort, not ex­ceeding either the price, or quantity propounded?

This que­stion

differs not from the former, ha­ving here connext a greater with a lesse, ex­cepting one, which is e­quall unto the price propounded, and might have been con­nexed unto a greater, but for one cause which shall be spoken of hereafter: so now to proceed in the operation of this Example, which is divided into severall columns: In the first stands onely the price propounded, viz: 5 s. In the second is placed each particular price: as 1 s. 3 s. 5 s. 8 s. 11 s. 14 s. with the least, as 1 s. I here connex 14 s. 11 s. & 5 s. & 3 s. is coupled with 8 s. this done, finde their diffe­rences betwixt these particulars and the price set, which transcribe according unto the former Exam­ples, so against 1 s. the difference will stand 9, 6, 0, against 3 will be 3. and against 5 stands 4 &c. finde the totall of these differences which is here 32. Then [Page 245] say by the Rule of proportion, as 32 is to 168 pound (the quantity to be mixt) so will 15 be to 78 ¾ pound. And thus repeating the Rule of Three so often as there be differences, the respective quan­tities to every one will be produced, as in the last co­lumne, whose totall is 168 pound, the first quantity propounded.

In most questions of this kinde, having found any one number, the rest may be discovered without re­petition of the Rule of Proportion: for by the Rule of Practice, Parag: 9. having found the difference and quantity for A or F, as admit F 21 pound, then must E & C have likewise the same quantity, their differences being equall, and D. 2 must be halfe so much, viz: 10 ½ lb, that is 10 lb 8 ℥, once that and ½ shall be the quantity for B 15 lb 12 ℥, and A be­ing 5 times the difference of B, the quantity of that sort for A must be 78 lb 12 ℥, the totalls as before, equall unto the quantity of the composition pro­pounded: and 168 Crowns is equall to 42 L. equi­valent to the particulars: for A 78 lb 12 ℥, at 1 s. the lb, comes unto 3 L 18 s 9 d. B 2 L. 7 s. 3 d. next C 5 L. 5 s. then D 4 L. 4 s. Fiftly, E 11 L. 11 s. Last­ly, F comes to in money 14 L. 14 s. the totall summe in money 42 L. as by the common price, and the quantity given.

The same Question varied thus, with the former prices, and quantity propounded.

In this Ex­ample

there is the same cōmon price w^th the par­ticulars, and quantity to be mixt, w^ch are thus alli­gated, viz: 1 s. & 14 s. next 11 s. & 5 s. also 8 s. & 3 s. these thus connext, and their differences transcribed (or counterchang'd accordingly as this rule requires) the summe of those differences will be 24: so the proportion is as 24 to 168 lb, so 9 unto 63 lb, the quantity of that sort to be taken: and so proceed according to the last Ex­ample you will finde the rest, as 21 lb 42 lb. 14 lb. 0 lb. & 28 lb. whose totall is 168 lb. as before: a­gainst these stands the value of them, according to each particular price, and quantity of that simple, whose totall is 42 L. as in the fi [...]st example; yet the state of the question is not performed according to demand, here being no part in the composition of that simple (represented by E) of 11 s the lb, as was in the last Example, which was the cause, it was con­next as before; and a caution to you hereafter, in any question of this kinde.

These Rules prescribed I hope may satisfie the Ingenious: yet severall Arts have various proposi­tions, [Page 247] one of most note is in the commixing of met­tals, in which there is a losse in melting, except in Gold or Silver refined from the faces, besides an alloy is usually allowed, therefore I will render you one Example of that kinde.

Question 7. A Gold-smith hath severall masses, or ingots of Gold, one purer than another; of 3 sorts he was to make a mixture, whereof one was 16 Carects: the second 18 Carects: the third was 20 Carects fine: this the Gold-smith was to mix with such an aloy, as that the whole mixture of 135 Ounces should be 14 Carects fine: How much must be taken of each sort?

First you

are to under­stand, that an ounce of gold is divided into 24 parts, cal­led Carects, and an Ounce of silver con­sists of 20 parts, each called a Penny-weight: to distinguish the finenesse of mettals, those that will indure the triall of fire, and not diminish in its weight, are said to be 24 Carects fine: if it loseth 2 Carects, that sort is said to be 22 fine: if it loseth in its triall 4 Carects then it is said to be 20 Carects fine, and so still termed in what remains of 24. and thus silver is valued in Ounces, as the gold is in Carects: for a [Page 248] pound of Silver tried, and loseth nothing, is term­ed 12 Ounces fine: if it loseth by fire 10 penny-weight, then it is said to be 11 Ounces, 10 p. fine: if it will not stand the triall, but loseth 1 Oun: 12 p. weight, then such a sort will be termed 10 Oun: 8 p. fine: and so for any other sort denomi­nated by the remainder. Lastly, an Alay or Alloy is nothing but a mixture of some baser metall, as Brasse, or Copper to adulterate the Gold, or abate the finenesse of it, and so likewise of Silver. And now for the solution of any question in this kinde; set down first the rate propounded and then the particulars under one another, as before; and for the Alay unknown, place a cypher under all: with which connex all the others, and counterchange the differences, which are all alike, and equall to the rate propounded, excepting the Alay, whose difference is 12, in this Example: and every one of the others 14; so at two Rules of proportion they are all found: for as the totall of differences 54, unto 135 ℥: so 12, unto 30 ℥ for the Alay: Again as 54 to 135 ℥, so 14 unto 35 ℥, and so many Ounces must there be of every sort of the Gold, noted with A, B, C, the Alay with the aspiration H; not that any one aspires to drosse in Gold; but Princes, and their People generally in respect of the World, whose in­fatuated Children, doe think it is all Gold that gli­sters.

This last question differs from all the former in two things; the one is, the compositions or mix­tures of them are made out of the simples, or par­ticulars given, and so the common rate, or price, is alwayes lesse than the greatest, and greater than the [Page 249] least propounded: Secondly, those may be varied in their alligations, whereas all questions of this na­ture admit of no connexion, but with the Alay; in the operation, and in other things, it differs not from the former, the totall difference being the first number, the quantity given the second; and each particular correspondent difference the third num­ber; and the summe of the fourth numbers found, must be equall unto the quantity propounded, if the operation be true, which is the triall of it.

If Gold of severall finenesse were to be mixt to­gether, without any Alay, but the quantity to be mixt known, and each particular difference, and the rate, or finenesse of the whole composition to be better than the worst sort, yet not so fine as the best, the quantity of each particular will then be found by the former prescribed Rules: so here I will con­clude this Paragraph, but not the subject. Having writ nothing as yet concerning the Apothecarie, which shall be the next: but not to involve his Sim­ples in conjuring words, to circumscribe Gramma­rians and charm the spirits of the Learned with com­mon compositions, as Children disguised with vi­sards doe amaze the Wise and Valiant: but here I will only shew how their Simples may be compoun­ded according to their qualities, whether Hot, Cold, Dry, or Moyst, in sympathy with the Elements, distin­guished in 4 Degrees, and how to make a Medium with either of the two extremes.

Question 1. Out of two Simples, to make a composition; which Me­dicine shall be of any mean temperature between the simples propounded, and of any quantity assigned.

First you are to

understand that the weights used by the Apotheca­ries are these, & commonly thus noted: a pound lb, containing 12 ℥ Ounces: an ounce 8 ʒ Drams: a dram 3 ℈ Scru­ples: a scruple 20 Graines: this done, admit a com­position to be made out of two Simples, one hot & drie in the fourth degree, the other cold and moyst in the third degree, the composition to be tempe­rate, and the weight of this mixture 1 lb or 12 ℥. Look in the Table for these degrees, where against 4 you will finde the Index to be 9 I, and for that which is cold and moyst in the third degree you will finde at B in the next columne the Index 2, and 5 the medium, or mean temperature for the composi­tion; which first set down, and then either Index, viz: 9 & 2, then finde the difference betwixt them, & the medium 5 which is here 3 & 4: these must be transcribed with each Index connext as in the former Examples: this done, the proportion will be as the summe of the difference, viz: 7, shall be in propor­tion unto the quantity to be mixt, that is 12 ℥, so shall each particular correspondent difference be [Page 253] unto their respective quantities to be taken in the Composition; which you may write (as they are found) against each Index, or proper letter; and these proportions must be repeated so often as there be particular differences: as in the bottome of the Table, noted the first and second Rule, whose totall in the third row is 12 ℥, the quantity given, which shews the work is true. The Simple which was hot and drie, is least in the Composition, being in the greatest excesse, and is here 5 ℥ 1 1/7 ʒ. the other B, cold and moyst, but in the third degree 6 ℥ 6 6/7 ʒ, the summe 1 lb. This Rule (and likewise any in Al­ligation) may be also tried by the 16 Axiome, Parag: 7. by the former Rules in mixtures of Sim­ples.

Question 2. A Composition is required of 3 Simples, whose quali­ties are knowne, and a temperature betwen one of those three is demanded.

The tempe­rature

of this Composition is required in the first degree of Cold & Moyst, whose Index is 4: the qualities of the Simples are these, the first, 4 degrees Cold and Moyst, whose Index is an unite noted with the letter A. The second Simple is Hot and Drie in the third degree, whose Index [Page 254] is 8 H. The third Simple is Hot and Drie in the fourth degree, noted with I, and the Index 9. there being two qualities greater than the temperature assigned, connex them both with 1, and finde the differences, as in former Examples, which proves here I 3, H 3, A 5 & 4, that is 9. the quantities of these Simples to be mixt, is 10 ℥ propounded. so the proportion will be (in the second row of this Table) as the summe of the differences 15, is to the quan­tity of the Composition, viz: 10 ℥. so 3 the diffe­rence is unto 2 ℥ for I. then H must have the same quantity, and A 6 ℥, as in the third row, the totall of these is 10 ℥, equall to the quantity propound­ed, and may be also thus tried, by the 16 Axiome, Parag: 7. multiply each difference by its respective Index as it stands, as 9 by 3, and 8 by 3. And thirdly, 1 times 9, the summe of these is 60. so will the pro­duct be made of the temperature assigned, as in this 4, and the summe of all the differences 15, which produceth also 60 as before, either way is triall suf­ficient: yet convenient it is to know them both, our wayes being doubtfull, and Man prone to erre.

Question 3. A Composition of 4 Simples is required whose qualities are known, and the quantity of any one is to be mixt with such quantities of the rest, that the quantity of the Composition may be of any temperature re­quired, greater than the least, and lesser than the greater given.

Admit the temperature of this Confection were to be Hot and Drie in the first degree, the Index to [Page 255] it will be found in

the former Table 6. the first Simple Hot and Drie in the fourth degree, whose Index is 9; of this there must be 6 ʒ added in the composition to the other three, whose qualities are these, G Hot and Drie in the second degree, whose Index is 7. D Cold and Moyst in the first degree, the Index 4. Lastly, B Cold & Moyst in the third degree whose Index is 2. the 4 Indices must be coupled, a greater with a lesse than 6, the Index unto the tem­perature required; in this, 9 is connext with 4, and 7 with 2. the differences are 2, 4, 3, 1, noted with these letters I, G, D, B, the quantity of I is here given 6 ʒ, whose Index is 9, and the difference belonging unto that (as counterchanged with another) is 2, which must be the Divisor, in questions of this na­ture, the quantity given the second number (in this 6 ʒ) and each respective difference the third num­ber in the Rule of 3, excepting that which apper­tains to the quantity given, as in the three Rules of the Table, viz: As 2 unto 6 ʒ, so 4 to 12 ʒ; for the quantity of the Simple belonging to G; which write in its proper place, and so proceed to the rest, where you will finde under the title of ʒ the summe of 30. or 3 ℥ 6 ʒ: this question having no quantity assigned in the whole Composition, is proved as was the last: that is, each difference multiplied by its re­spective [Page 256] Index, whose totall will be equall (if the operation be true) unto the summe of the differen­ces, multiplied by the Index of the degree given: as here 6 by 10 produceth 60; and so is the summe of their severall products, as in the triall of the Table appears, whose quantity proves 30 ʒ, or 3 ¾ ℥ in the temperature of Hot and Drie, and in the first degree as was desired.

Question 4. A Composition being made of divers Simples, whose qualities and quantities are certainly known; and it is required, in what degree of temperature this Confection is in?

In all que­stions

of this kinde, there is no more to doe than first to set downe each Index, according to the temperature of the Medi­cine assigned: against each quality place the quan­tity of it, which multiplied by its respective Index, and the summe of those products divided by the to­tall of the quantities, the quotient will be the Index unto the temperature of the whole Composition. As for Example, there is a Confection made of 4 Simples, whose qualities and quantities are as fol­loweth: 1 ℥ temperate, whose Index is 5. 2 ℥ hot and drie in the first degree, whose Index is 6. Then 3 ℥ cold and moyst in the third degree, whose In­dex [Page 257] is 2. Lastly, 4 ℥ in the second degree cold and moyst, whose Index is 3: the products of these are, 5, 12, 6, 12, the summe of these is 35; the totall of the quantities is 10 ℥, with which divide 35, the quotient is 3 ½ for the Index required, which in the former Table (of degrees and qualities) falls be­tween C & D subtract the Index 3 ½ from 5 the medium, the remainder will be 1 ½ Cold & Moyst, the question solved: if the Index had been greater than 5. subtract then 5 from it, the remainder will be your desire, in the degrees of Hot and Drie. This question depends upon the common Rule of Pro­portion, viz: As an unite is to the Index, so will the quantity be unto a fourth proportionall number, as in this last Example does appear.

Question 5. A Confection to be made of severall Simples, whose particular quantities and qualities are known, as in respect of any degree in Heat, Drought, Cold, or Moysture, how to finde the temperature of such a composition.

In Compositi­ons

of this na­ture set down the quantity propoū ­ded twice, as in these Tables of Hot & Cold, with the other two ex­tremes Drie and Moyst: and according to the de­gree of the Simples temperature, write down the [Page 258] Index, which uultiply by each particular respective quantity, the totall of the products divided by the summe of the quantities, will shew the Index for the temperature required: as in this Composition, whose qualities and quantities are these, one Simple in weight 5 ℥ cold in 4 degrees, whose Index is 1, and moyst in the first degree, the Index 4. then 4 ℥ cold in the second degree, the Index 3. and drie in the fourth degree the Index 9. then was there taken 2 ℥ of a Simple in the qualities temperate, and mixt with 3 ℥ of a Simple hot in three degrees, and drie in two, the Indices 8 & 7. Lastly, 1 ℥ of a Simple hot in the fourth degree, and moyst in the second degree, the Index to these is 9 & 3. these multiplied by their quantities produceth 5, 12, 10, 24, 9, and 20, 36, 10, 21. 3. their totalls 60 & 90 these divi­ded by 15, the quotients will be 4 & 6, each an In­dex to the quality of the Composition required, viz: hot in one degree, and in the first degree of moy­sture, the question solved.

Question 6. To encrease or diminish in quality any composition or medicine, according to any degree of temperature that shall be assigned.

Admit the Com­position [...] or Me­dicine given w^ch was hot in the first degree, and moist in the first, as in the last que­stion; and sup­pose [Page 259] that 6 ʒ was the quantity of the Medicine, which by another confection is to be encreased or diminished in either of the qualities, to any degree assigned, as here, the quality which was hot in the first degree, is to be encreased unto the third degree of heat; or that in the first degree of moysture, to be made temperate. In this Example there are two Tables, the first is in one degree of heat, whose In­dex is 6. yet by the commixture of another Simple is to be encreased unto the third degree of heat. Take any Simple or Composition, which is either equall in the temperature, or greater, in this I take one in the fourth degree of heat, whose Index is 9: place this under the Index, whose temperature is known, viz: 6. the difference of these two, and 8 propounded is counterchanged 1 & 2, then say, as the difference of that known, and that given, or the difference of the Index 1 propounded to the quan­tity of the Confection made 6 ʒ; so shall the diffe­rence of 2 (the Index) be in proportion unto the quantity of the Simple that is to be mixt: so it is, as 1 unto 2, so 6 ʒ to 12 ʒ; or as 1 to 6 ʒ, so 2 unto 12 ʒ. And so much of that which is hot in 4 degrees, commixt with another hot in the first de­gree, the temperature will be in three degrees of heat, as will appear in the Table of their qualities. And so likewise in the second Table of this Example, the Confection which was moyst in the first degree is to be made temperate: D 4 is the Index unto the quality given, which must be annihilated, like the common people of these times, as neither hot nor cold. Take the Index of some required Simple, whose degree is either equall, or greater: as here [Page 260] admit one, in the second degree of heat, whose In­dex is 7 G. the difference of each Index transcribed according to the last, or first Question of this Parag: the difference D 2. & G 1. now the proportion will be, as 2 to 1 so 6 ʒ unto 3 ʒ, or 2 unto 6 ʒ: so 1 to 3 ʒ. and 3 Drams of a Simple, hot in the second degree, added to a Confection of 6 Drams, moyst in the first degree, in respect of moysture it would have been in that quality, neither hot nor cold, nor yet drie or moyst, if their qualities correspond.

In the first of these Examples, the Composition 6 ʒ, hot in the first degree, being mixt with 12 ʒ hot in the fourth degree, the whole Confection was made hot in the third degree, but yet is held moyst in the first degree, as it was before: although some affirm, not onely heat does abate cold, drought & moysture, but also moysture and drought lessens cold; two of these qualities being but by accident. So to return, if the last Composition which was made hot in the third degree, and moyst in the first, the whole Confection of 18 ʒ, may be encreased, or diminished in the degree of moysture by one of the last Examples, to which I referre you, repetiti­ons being unnecessary: if the quantities of any Simple, or Composition to be made, should happen in a fraction, or in severall denominations, viz: lb ℥ ʒ ℈ or the like: reduce the given quantity into the least denomination, and then proceed as be­fore: if there happen a fraction in the Index, sub­tract it from 5, or the Index of the meane tempera­ture from that, and you will finde your desire: as admit the Index were 3 ½ subtracted from 5, shewes the degree 1 ½ cold and moyst: if the Index were [Page 261] 8 ¾, the degree appertaining to it (by the subtracti­on of 5) will prove 3 ¾ hot and drie, and so for any other fraction in these degrees and qualities; of this subject I will write no more, lest the Apothecaries should take me for a simple, or this the worst of their drugs.

Question 1. There was a man had two sons and one daughter, viz: A, B, & C, the old man (in his last Will & Testa­ment) bequeathed to his children all his estate (in value unknown) in this manner: to his eldest son A, he gave a portion double to B, his second son; and his part, treble his daughters C: the estate after his death was praised at 745 L. What must the chil­drens parts be?

Most Rules of single position consist of more parts, or proportionall numbers than one, and usu­allly [Page 263] differs not

in form, from the Rules of So­cietie, nor yet in the manner of operation: in these questions take such numbers as may be pro­portionable, or answer the state of the question, and yet generally avoiding all fractions: first set down the names, or the thing, upon which the demand is made, and those underneath one another, as here A, B, C, take commonly the least number, or part, and in the least denomination; as for C, I suppose 1. then B must be 3 times so much, against which I place 3. the portion of A must be twice that, for which I put 6, the summe of these is 10. then by the 15 Axiome, Parag: 7. the proportion will be, As the summe of the supposed numbers 10 shall be in proportion to the totall summe of the Legacies given, so will each particular or proportionall part be to his respective share or portion: and this Rule of proportion must be repeated so often as there be supposed parts for the required shares, that is, As 10 unto 745 L. so will 6 A be to 447 L and in the same manner proceeding you will finde for B 223 L. 10 s. and for C the Daughters portion 74 L. 10 s. the totall 745 L. which shewes the worke is true. Many times in these questions you may ease your selfe in the operation, as here, 1 being a Multiplier, and 10 the Divider; by divers former Rules there needs no other division than cutting off the first fi­gure on the right hand, so it will be 74 5/10, that is, 74 L. 10 s. for C three times, that is, B 223 L. 10 s. twice, that is, 447 L. for A, as before.

Question 2. Five Merchants, as A, B, C, D, E, entred into So­ciety with severall stocks, and according to their shares for to stand the pleasure of Fortune either in gaine or losse: profit was their hopes, and proved the end: their stock unknowne, the gaine 675 L. their conditions were, that ½ the gaines of A, should be ¼ part of B. and ¼ part of B, should be ⅕ part of C. three times A the gaines of D. and twice what B did gaine, was the gaines of E, each particular share is required.

The conditions

here being onely of their gaine or losse, you may take any propor­tional numbers in this case; but such as will solve the question without fractions, is the best, for which you are not confined to any particular number, as 2, 4, 5, &c. herein I doe take the least, and suppose 2 for the gaines of A; since halfe that is to be the fourth part of B, in this proportion, B must be 4, and C 5, being five times the fourth part that B did gain: then D must be 6, being three times A. and E will gain 8, being twice the adventure of B. these supposed numbers, being placed under one another, (against each particular profit) finde the summe of them, as in the Table 25. which must be the first number, the whole gaines, or losse the second, in [Page 265] this 675 L. so will 2, 4, 5, 6, 8, each particular sup­posed number (according to the 15 Axiome, Parag: 7.) be in proportion to their respective shares, so A must have 54 L. B 108 L. C 135 L. D 162 & E 216 L. the summe of these 675 L. the whole gaines, which is proof sufficient, as by the Table appears. The common multiplier 675 L. with the total of the supposed parts; viz: 25, the Divider unto all may be reduced by 5 or 25 (according to the former Rules of Practice, Lib: 2. Parag: 9. Quest: 3.) unto this, as 1 unto 27, so each supposed number to its particular part; by which meanes a division is a­voided in all.

Question 3. The summe of 570 L. was to be distributed unto 3 men, A, B, C; but at severall payments, and upon these conditions, that when A received 2 L then B should have 4 L. and so often as B took 3 L. was C to have 5 L. in this proportion was the money payd, and here their shares are required.

The Table of

this Example is di­vided into 5 rows, as by the columne in the head ap­peares: the state of the question is, where A had 2 L. B must have 4 L. and in that pro­portion, as if B had 3 L. then C should have 5 L. in the first row they stand, as A 2. so B 4: and for B 3 then C must have 5. in all such cases proportionall parts may be found in whole numbers thus: multi­ply [Page 266] A 2 by B 3, the product is 6 for A. then multi­ply B 3 by B 4, the result will be 12 for B. and 4 thus encreased by 5, produceth 20; that is B by C, so these three numbers are A 6. secondly, B 12, & C 20, which are proportionall to the state of the question, and the thing required: for B 12 is in a duplicate proportion to A 6. as A 2 was to B 4. and so 12 is to 20. as B 3 is unto C 5. so in the second row the three proportionall numbers are A 6. B 12. & C 20. which you may reduce if you please, as in the third row unto 3, 6, 10, the summe of them, as in the fourth row is 19; this must be the first num­ber; the quantity or summe, as 570 L. the second, the other the parts of A 3, B 6, & C 10, according to the 1 or 2 Quest: of this Parag: and should be pla­ced in the same manner, the operation as in the fift row where the share for A is 90 L. and so finde the rest as B 180 L. and the part for C 300 L. the totall of them 570 L. as before; each share being in pro­portion, unto the state of the Question propounded.

Question 4. There came unto a wedding 6 Lords, 8 Knights, 24 Esquires, and 48 Burgesses: these gave at an offer­ing 120 L. in this manner, viz: 2 Lords gave as much as 4 Knights, 5 Knights equall unto 6 E­squires, and 5 Esquires gave so much as 10 Bur-Burgesses. How much did each distinct degree of­fer?

In all questions of this nature you must first finde out proportionall numbers answering the state of the question propounded, which to effect, according unto the last Example, set downe the proportions [Page 267]

given, as in the Table to this Question, where the offering of 2 L, 5 K, & 5 E are multiplied together, whose product is 50. then 5 K, 5 E, & 4 K, is 100. next 5 E, 4 K, & 6 E 120. Lastly, 10 B, 6 E, & 4 K, which multiplied produceth 240. the summes of these are 50, 100, 120, 240, and reduced unto their least denominations, 5, 10, 12, 24, which are pro­portionall numbers, and manifesteth that 5 Lords offered equally unto 10 Knights, 12 Esquires, or 24 Burgesses.

These proportions

being found, place them in order, as here you see: then suppose for 1 Lords offering any summe you please, admit 20 s. then 6 Lords offered 6 L. which set down; the proportionall numbers found were 5, 10, 12, 24, and according to the supposi­tion, 5 L. answers unto any of the proportionall numbers, by which finde the rest, as 10 to 5, so 8 unto 4 L. the supposed part for 8 Knights. and in this manner finde the rest, viz: as 12 to 5 L. so 24 unto 10 L. and as 24 unto 5 L. so 48 unto 10 L. [Page 268] the summe of all the offerings (according unto sup­position) amounts unto the summe of 30 L.

This summe discovered by the supposition is but 30 L. and the offering was certainly known 120 L. therefore institute the Rule of Proportion again, viz: as 30 to 120 L. so 6 unto 24 L. and so much did the 6 Lords offer, and so repeating the Rule you will finde the 8 Knights offer 16 L. the 24 Esquires 40 L. and likewise the 48 Burgesses offered 40 L. the totall is 120 L. the summe propounded, and the particular parts required.

Question 1. There was an Excise upon all goods, sold by whole-sale men: and by an Edict, if any man rendred not a true account, the penalty (if discovered) was double the commodity: yet all trades-men allowed to be as obscure as they could, the truth being affirmed: it was demanded of one man, what he had sold: Who replied, 2 pieces of Canvas, 2 of Fustian, and 7 of Holland, every piece a crown more than other; and one piece of the best, was 3 times the price of the worst: what was the value of the 11 pieces of cloth?

In this Proposition I take the last Example, in which I choose the meanest price, and make two sup­positions thereof, and supposing the price of that piece to be 8 crowns, and being there were 10 p e­ces more, every one dearer by 5 shillings, the best then must consequently be 18 crowns, which accor­ding [Page 274] to the supposition, should contain the worst three times, whereas here 3 times 8 is 24, from whence take 18, the remainder is 6 for the first er­ror, and too much: for the second supposition you may take a lesse, but here I will againe suppose a greater, as 12 crowns, and being it is of the same denomination, the best piece should be worth 22 crowns, and that equall unto three times the suppo­sition; but 3 times 12 is 36, from whence take 22, the remainder will be 14 for the second errour, which place under 12 the supposition with the signe of more: this second error, multiplied by 8 the first supposition, produceth 112: and 6 the first error, multiplied by 12 the second supposition; the pro­duct will be 72: the difference betwixt these pro­ducts is 40, for the dividend, the difference of er­rors 8, the divisor; the quotient 5, for the true number required, and then the best piece must be 15 crowns; being 3 times the value of the other, the prices of all the rest will be easily discovered, as thus:

Two pieces of Canvas, the first 1 L. 5 s. the other 1 L. 10 s. then two pieces of Fustian, the first 5 s. more than the last piece of Canvas, viz: 1 L. 15 s. the other 2 L. the piece, then for the 7 pieces of Hol­land, the first 2 L. 5 s. the second 2 L. 10 s. the third 2 L. 15 s. the fourth 3 L. the fift 3 L. 5 s. the sixt 3 L. 10. the last 3 L. 15. that is 15 crownes, and three times the first piece in value, according unto the state of the question, the totall summe is 27 L. 10 s. the thing required: this is a sufficient triall; and yet for to please all, according unto the former Rule, as 8 is to 6, so will 4 be to 3, which fourth [Page 275] proportionall number, taken from 8 (the first sup­position) the remainder will be 5. the number re­quired: the errors being both in excesse.

Question 2. To divide any number propounded into any parts that shall be required; and those parts for to be in any proportion one unto another, that shall be assigned.

The given number here

is 45 for to be divided in­to 2 parts, and those to be in a triple proportion one unto the other: the Crosse made as before, let 8 be the first supposi­tion, and the least part, then 24 should have been the greatest, the summe of them is 32, but the num­ber given is 45, therefore this supposition is 13 de­fective, as by the error does appear: then suppose 10 for the number required, the triple of it is 30, the summe of them both is 40, which shews the se­cond error for to be 5 defective: these errors mul­tiplied crossewise, into their supposed numbers, will produce 40 & 130, their difference 90 for the Di­vidend: the difference of Errors 8 the Divisor, the quotient 11 ¼ for the lesser number required, three times that is 33 ¾, their totall 45, the question sol­ved. By the first Rule, the proportion is as 8 to 13, so 2 unto 3 2/8 or ¼, which 3 ¼ (according to the first Rule) must be added unto 8 the first supposition, the summe will be 11 ¼ as before, answering the state [Page 276] of the question, for the lesser number which being discovered, the other is easily found; this question will be performed by the Rules of single position, if you take proportionall numbers, answering the state of the proposition.

Question 3. To divide a given number into any two parts, and those in any quantity assigned, as to part 10 in two; and so, as that the greater divided by the lesse, the quo­tient shall be 20.

In all questions of this

kinde, suppose any one number, the other is the remainder; as here I sup­pose 2: the other must be 8, both numbers be­ing 10: and according to the state of the que­stion, the greater should contain the lesse 20 times, then consequently 20 times the lesse would be equall unto the greater number, so 2 multiplied by 20 produceth 40, and should have been equall to 8, the error is 32 too much, then take a second supposition, as admit 1, the other part must be 9, and the second error 11: and both too much, which note, and then multiply them into their contrary suppositions, the products will be 22 & 32, the difference 10 for the Divi­dend, the difference of errors 21 the Divisor, the Quotient 10/21 for the lesser part, which subtracted from 10 the remainder will be 9 11/21, which is 20 [Page 277] times the other, and consequently 9 11/21 or [...]00/21 divi­ded by 10/21 the Quotient will be 20, the thing requi­red: and for triall, by the first Rule of False posi­tions, as 21 to 32, so 1 unto 1 11/21; which (according to the same Rule) subtracted from 2, the first sup­position, the remainder will be 10/21 for the true num­ber as before.

Question 4. A vessell of 63 gallons was filled with French wine of two sorts; the one was at 2 s. the gallon, the other at 2 s. 6 d. the gallon; the wine in the hogs-head thus filled, did come unto in money 7 L. 4 s. and it is here demanded how much there was of either sort.

The quality here, and

the number of gallons in either supposition, will solve this question, the denominations being the same in both, as in respect of the quality, that is, sup­posing the best, or worst sort, or price in either; yet to avoid any great num­ber, the quantity of gallons being odde as 63, and the meanest price or quality being even, I take that in both, and presuppose 23 gallons of the meanest sort, then there was 40 gallons of the best, which at 2 s. 6 d. the gallon comes unto 5 L. and the other sort unto 2 L 6 s. in all 7 L 6 s. the price was 7 L. 4 s. from whence it is apparent, that the first error was [...] s. too much [...] then suppose 33 gallons (or what you please) which granted, there must be 30 gallons [Page 278] of the best, which comes unto 3 L. 15 s. and 33 gal­lons of the worst (at 2 s. the gallon) amounts unto 3 L. 6 s. the summe 7 L. 1 s. the error 3 s. defective; these errors multiplied crossewise by the suppositi­ons, will produce 69 & 66: and according to my former directions (being the signes are unlike) their summe is 135 for the Dividend, the summe of the errors 5 for the Divisor, the Quotient 27, the num­ber of gallons, of the worst sort of Wine, then must there be 36 gallons of the best, the totall quantity 63 gallons; the price of the worst is 2 L. 14 s. and the best comes unto 4 L. 10 s. the just summe of 7 L. 4 s. according unto the proposition. And by the se­cond Rule, the proportion is, as 5 unto 2, so 10 to 4, which 4 if it be added unto 23 (the first supposi­tion) the summe will be 27 gallons, as by the for­mer, operation in the figure does appear.

Question 5. Hiero King of Sicylia, caused a Crown of Gold for to be made in weight 10 lb. and it was conceived, that the Work-man had put a great Allay of Silver unto it, which abuse of the Artificer Archimedes de­tected; and by False positions may be thus dis­covered.

The making of this

Crown is mentioned by Vitruvius, and others; but by what Rule Archi­medes discovered it, is re­maining in obscurity, on­ly that the conclusion he found out by water; the quantity of gold, and the weights then used at Sy­racusa, as uncertainly known as the rest; therefore here I will onely state a Question of the same nature, by which the Artificers dishonesty, either was, or might have been thus by art detected; It is imagined that Archimedes had of the King a masse of gold equall in weight to the Crown, which here I suppose might weigh 10 lb. he made also another equall masse of silver, with which mettall the gold was alayd, but in what quantity unknown, but this gran­ted; it might be thus found: first there was provi­ded a cisterne, and that filled with water up to a spout, under which there was another vessel for re­ceiving the overplus of water.

Admit by putting the Crown into the cistern of water, the quantity run out was 2 ⅕ lb. the masse of pure gold avoided but 2 lb water, and that of silver 3 lb. which in the first place shewed the difference of mettalls, for they being of equall weight, that most compacted and heaviest of nature will have the lesser body, and consequently possesse the lesser room: by this a great allay appears, and will be ex­plicitly known, as thus, suppose there was 4 lb of silver, then was there 6 lb of gold: here institute the Rule of proportion twice for either mettall, as in [Page 280] the first Table, viz: if 10 lb of gold voyded 2 lb of water, how much will 6 lb of gold voyd, which will prove 1 ⅕ lb of water; and so according to the sup­position finde how much water the silver will a­voyd, which is here also 1 ⅕ lb, the summe is 2 ⅕ whereas the water which the Crown forced out was 2 ⅗ lb. the difference onely ⅕ lb for the first error. then suppose there might be 3 lb of silver in the Crown, there must be of gold 7 lb. and according to the second Table the fourth proportionall num­ber will be 14/10, or 7/5. then again, as 10 s. to 3 W. so 3 lb silver to 9/10 lb of water, the summe of these is 2 3/10 lb water, the difference of this 2 3/ [...]0 and 2 ⅕ is 1/10 too much for the second error. These multiplied by their contrary suppositions, will produce ⅖ & ⅗, and according to my former directions, the diffe­rence is ⅕ for the Dividend, the difference of errors 1/10, the Quotient 2 lb of silver, the quantity of the alay, and 8 lb the weight of gold, that was in the Crown, the thing required.