Question IV.

If there be Ten Perfons of Four feveral Countries, English, Dutch, French and Spaniards, to pay a Debt of cool. fo that every English-man pays 501. every French-man 701. every Dutch man 1301. and every Spaniard 150 1. How many is there of each Country?

T

HE Dividend (according to the former Rule) is 500: Now to find the Divifor, take his Sum that pays leaft (namely 50 ) out of each of the other three, 150, 130, and 70, and the remains will be 100, 80 and 20.

And the First and Last, viz. 120 for the Divifor; the Quotient will be 4 and the Spaniards cannot be more.

2

12

Secondly, Add the firft and fecond together, viz. 180, the Quotient is 2; and the Spaniards cannot be lefs. That is,

The Spaniards cannot be much more than 4, or lefs than 2: And therefore, feeing any one Solution will ferve,

Let the Spaniards be 3, and by that multiply 100, and take the Product out of 500, there remains 200 for a fecond Dividend, which divided by the fecond remain, 80, the Quotient is 2, therefore the Dutchmen are 2, which multiplied by 80, makes 160; take that out of 200, there remains 40 for a third Dividend, which divided by the third remain, 20, the Quotient is 2 for the French-men alfo, and confequently the English-men must be 3, because all of them are 10: But the Spaniards may be alfo 4 or 2.

The Reason why the Spaniards and English, as alfo the Dutch and French, are equal in number, is, because their Payments differ equally from 100, which is the Mean Sum with which 10 Men fhould pay 1000 Pound.

Question

Question V.

If one should buy 12 Loaves of Bread for 12 Pence; fo that fome might be Twopenny, fome Penny, fome Halfpenny, and fome Farthing Loaves; and it be required to know how many he must buy of each fort?

NOW, Because of 12 Loaves for 12 Pence, the Mean Price is 1, but one of the Particulars being alfo 1, there fhould be no Penny Loaves, because there is no difference betwixt the Mean Price and One Penny.

But it may be found by this Rule to be either

If any Perfon write down any three Digits (as 6, 8, 7) and under them make Nine other Digits fet in Rank and File; and under them you fet Nine other Digits in the fame order; how to know, and fet down the Agregate or total Sum of all the 18 Digits and the three firft Figures, being added together; before any of the 18 Digits be fet down.

687

ET the three Figures first fet down be 6, 8, 7, under them draw a Line, and under the Line make nine Pricks; and under them draw another Line, and under it make nine other Pricks, and a Line under them, as is done in the Margin: This done (always) fubftract 3 from the Digit ftanding in the place of Unity, in this cafe 7, and the remainder is 4; then, upon a piece of Paper (by the bye) write down 6,8,4 inftead of 6, 8, 7, and to the left hand of it fet 3, inftead of the 3 which you abated from the 7; fo will your Number be 3684; and that will be the Aggregate or Sum of the 18 Digits, and the three uppermoft Figures, all being added together: As in the following Example.

Having

Having fet down the three Figures 687, and drawn under 687 them a Line, and made 18 Points and Lines as in the Margin 748 above, bid any Person about you write any nine Digits upon 631 the nine upper Pricks, and you will write nine other upon the 254 nine other Pricks below; all which being added together, 251 fhall make 3684. Suppofe (as in the Margin) on the upper368 moft nine Pricks be written 748-631-254; then do you, 745 upon the nine Pricks under, write the Complements of those 3684 nine Figures to Nine; as for his three firft 748 do you write 251, for 631 write 368, and for 254 write added together, will make 3684, the Sum you first set down before any 745; all which of the 18 figures were written.

I

422 Example 2. But if the Digit in the place of Unity be less 325 than 3, as in these three figures 422, you must take 3 from 769 12, and there will remain 9, which fet down; and (because 134 you borrowed 10) take 1 from 2, and there remains 1, and 674 4 is the fame 4, before which put 3 for the 3 which you bor230 rowed, and then the Sum is 3419, which you tell beforehand 865 will be the Aggregate or Sum of all the Addition, as in the 3419 Margent is plain.

The like may be done by 4, 5, 6 figures, by obferving the fame method, by abating of fo many Unites from the place of Unity, and reftoring again in the place of Thousands or Ten thousands, &c. As in thefe Examples,

6785

25798

If Three of a fort of the Five Odd Digit Numbers be set in Rank and File, as in the Margent, and it be required of any Five of these Odd Digits to make the juft Number 20, How may that be done?

III In the Performance of this there is a Falacy; for 666 333 no five odd Numbers taken how foever, can make 555 up that Number; wherefore they do invert the 777 Numbers, by turning of the Paper upon which they 999 are written, upfide down, and then the three Nines become three Sixes; and fo 3 times 6 is 18, and two

of the Ones makes 20, as in the other Margent.

III

GEO

From a Point, in a Right Line given, to erect another Right Line which shall be Perpendicular to the Right Line given.

Definition I. J

A

Point is that which hath no Parts, and is the least
imaginary thing that can be conceived in the mind:
As this Point or Prick noted with Z.

Definition 2.] A Right Line is a Line drawn equally between two given
Points, and is the fhorteft diftance between them, as is this Line XY,
which is the fhorteft diftance between the two Points X and Y.
X
Y.

"

Definition 3.] A Right Line is faid to be Perpendicular to another Right Line, when it maketh the Angles on either fide of the erected Line equal; that is, fo that the erected Line inclines not either to the Right band, or to the Left, but ftandeth upright upon the Line from which it is erected: As in the Right Line A B, is faid to be Perpendi cular to the Right Line CD, upon which it is erected, for that it in clineth neither to the Right or Left hand; and because the Angles on either fide thereof are equal; namely, The Angle ABC on the one fide, equal to the Angle A B D on the other fide; either of which Angles are Right Angles, and the Right Line AB fo ftanding is Perpendicular to the Right Line C D upon which it is erected.

F

H

CD, and let it be required to erect another Right Line which shall be Perpendicular thereunto, from the Point B.

Open your Compaffes to any convenient small diftance, and fetting one Foot in the Point B, with the other make the two fmall marks E and F, on either fide, equidiftant from the given Point B.This done, Open the Compaffes again to any convenient diftance (greater than

the former)and fetting one Foot in the Point E,with the other defcribe the obfcure Arch GG (over the given Point B as near as you can guess). Again; (The Compaffes being ftill open at the fame distance) fet one Foot in the Point F, and with the other describe another obfcure Arch HH, croffing the former in the Point A: So is A a Point found, through which if you draw a Right Line from the given Point B, that Right Line A B, fhall be Perpendicular to the given Right Line CD, and from the Point B, which was required to be done: And the Angle A B D, on the one fide thereof, is equal to the Angle ABC, on the other fide; and both of them are Right (or Square) Angles.

Note, An Angle is always fignified by Three Letters, as a Point is by One; the middlemoft of which Three reprefenteth the Angular Point, as in this Cafe the Letter B. B being the Angular Point, and the Lines A B and B C the fides containing the Angle B.

PROB L. II..

How to erect a Perpendicular, when the given Point is in (or near 】 the end of the given Right Line.

Practice. ]

H

HERE are feveral ways to effect this; of which I will here fhew you only Two, as being the best.

TH

Let A B be a Line given, and from the Point A, towards the end thereof, let it be required to erect the Perpendicular A C. First, Open the Compaffes to any fmall distance, and fetting one Foot in the given Point A, with the other describe an Arch (or part) of a Circle, F ED, And (keep

ing