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Thus it it will be eafy to find out and collect all the limitted Answers to any Question of this kind, wherein there are only three Quantities propos'd to be mix'd: But when there are more than three, then the Work requires a little more Trouble; because the fingle Limits of all the Quantities above two must be found; that is, if there are four Quantities concern'd in the Question, the Limits of two of them must be found; if five Quantities are concern'd, then the Limits of three of them must be found, &e. as in the following Question.

Example 2. Question 18.

Suppofe it were required to mix four forts of Wine together, viz. One fort worth 7 s. 4 d. the Gallon, another fort worth 4s. 7d. the Gallon, a third fort worth 3 s. 8 d. the Gallon, and a fourth fort worth 2 s. 9 d. the Gallon. How many of each fort may bė taken to make a Mixture of 63 Gallons, fo as the whole Quantity may be fold for 5 s. 6 d. the Gallon, without Lofs, &c.

Firft let all the feveral Rates, and the mean Rate be reduc'd to one Denomination, viz, into Pence.

that

Viz. 7 s. 4 d. 88 d ; 4s. 7d. = 55 d ; 3 s. 8 d.=44d; 2s. 9d. =33d; and 5s. 6d. 66 d. Then put a the Quantity, of that worth 88d. the Gallon ;e that of 55d. the Gallon; y 444. the Gallon; and u that of 33d. the Gallon. Then 1|a+e+y+u63, by the Question.

of

and 288a55e+447 +33u63 × 66 4158. I a ze+y+u=63—2.

-

2-88a 455 2 =
55e +44 +33 4158-88a.

5 u=2079
33e +33 +33 = 207933 a.

3× 33

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-5622e+

22e11y2079-55a.

is;

2e+y=1895 a. Hence a is (37).

8
55 55e +55 + $5 % = 3465 — 55 a.
911y2233a - 693.

10y+24=3a63. Hence a 633 (21).

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From

Part XI. From the 7th and 10th Steps it appears, that the Quantity of. that fort of Wine denoted by a must be lefs then 37 Gallons, and greater than 21 Gallons: That is it may be a= any Number of Gallons betwixt 21 and 373.

Next to find the Limits of e, y and u.

2e+

Suppofe 11 a 22, then will 5 a 110, and 3a66. But 12 2e + y = 1895a79, per 7th Step. 12-2013 Y = 79 22. Hence.e (394).

Again

14y+u=63—a=41, per 3d Step.

1415y+u41 to

15-13|16|ue-38. Hence e 38.

From the 13th and 16th Steps it appears that if a 22 then e39, y = 79 2e = 1, and u — e —

Again,

38 = 1.

Suppofe 17 a 23, then 54 115, and 3a 69.
But 18 2ey 1895a74, per 7th Step.
2ey=189
18-2019 y = 74 -2e. Hence e742 (37).
Again 20e + ÿ + u =
u 63 a 40, per 3d Step.
e 21 y + u=40

20

-e.

21-1922 ue 34 Hence e 34.

From the 19th and 22d Steps it appears that if a 23, then e may be 35 or 36.

Once more for a further Illuftration.

Let | 23 a

24; then 5 a

5a==

120, and 3a=72.

=

69

But 24 2ey 1895 a 69, per 7th Step. 2426 25y = 69 2e. Hence e222 (34%). Again 26e + y + u = u = 63a39, per 3d Step. 26 27 ÿ+u=39—e,

27-25 28 u -e 30. Hence e 30.

From hence it appears that if a

24, then e may be either 31, 32, 33 or 34, viz. it may be any Number betwixt 30 and 34 by the 25th and 28th Steps: From whence the Values of y and u may be eafily found;

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Proceeding thus with all the other Values of a; there may be above 120 Anfwers found to this Queftion in whole Numbers, and if you pleafe to put a to Fractions, there may be found an innumerable Sett of Anfwers.

Those two Examples being well underfood (especially if the laft be thoroughly pursued) may fuffice to fhew the Method of Limitting Answers to all forts of Queftions of this kind.

Sec. 3. Questions producing Simple Quadzatick, &ca Equations.

Question 19.

A certain Footman A departed from London towards Lincoln ; and at the fame time another Footman B departed from Lincoln towards London, each keeping the fame Road. When they met A faid to B I find I have travell'd b (20) Miles more than you, and have gone as many Miles in c (6) Days, as you have gone Miles in all hitherto. But faid B at the end of d (15) Days hence I fhall be at London, if I travel ftill after the fame Rate. Quere the Distance of these two Cities from one another, and how ma ny Miles each Footman had travell'd when they met.

1. Suppofe the Number of Miles that A went each Day to be a, the Number of Days that he was on the Road to the time that he met B = z; then the Number of Miles that he went to the time he met B will beza.

2. Suppofe the Number of Miles that B went each Day -e ; then the Number of Days that B was on the Road to the time that he met A is (fince B parted from Lincoln at the fame time. that A parted from London); and the Number of Miles that he went to the time he met A will be ze.

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8,9z=√cd (=√6} × 15 = 100 10) Days,

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ze + b

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ze + b

N

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+40

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100) the Number of Miles that A and B went when they met ; that is the Distance in Miles from London to Lincoln.

Question 20.

It is is required to find three fuch Numbers, that the Sum of the First and Second Multiplyed by the Third, may be b (63); and the Sum of the Second and Third Multiplyed by the First, may be c (28); alfo the Sum of the First and Third being Multiplyed by the Second may be = d (55).

Let 14 and y reqrefent the First, Second and
e
I Third Numbers fought.

Then 2 ayey=b)

3 ea+ya=c per Question.
4aeyed)

2+3+4 52 ay + 2ey+2ae

2ay +2ey + 2aeb+c+d.

Let 6b+c+d=s.

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4aa

14 × 4aa 15 2 saa 4caa = ss - 2ds 2bs + 4b d.

35÷25—40|16|aa —

ss-2ds

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2bs + 4bd

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There is a Wall containing b (18225) Cubical Feet. The Height is c (5) times the Breadth; the Length is d (8) times the Height. I demand the Breadth of the Wall.

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{Wall.

Suppofe I
Then 2 ca is

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the Number of Feet in the Breadth of the

the Numb. of Feet in its Height dea the Numb. of Feet in its Length Queftion. jaxcaxdcadcca3b.

b

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5.36a= 3/

dec

Foot).

}

By the

(=8x5x=√91.125=4.5

CHA P. II.

Ductions producing Adfected Quadzatick. &c. Equa¿

15

Lemma.

tions.

A B C D, Sc.
1-1-1-1-

F along with the one thing A, in any ftraight Line, be plac'd any Number of Things B, C, D, &c. at any Distance or Distances from one another; I fay the Number of the things A, B, C, D, &c. is by one more than the Number of Intervals AB, BC, CD, &c.

Demonftration.

If in the faid ftraight Line along with, and at any Distances from the one thing A, you place things B, C, D, &c. ( being equal to any whole Number whatsoever) it is evident that you place z Intervals AB, BC, CD, &c. Confequently the Number of things is 2+1, and the Number of Intervals is n. Q.E.D.

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