ALL 



jiUl^allon is of two kinds, medial and alternate ; to w!»ich 

 fome add a third, caMed pjrliiil. 



Alligation medial, teaches how to find the mean rate of 

 a mixture, when the particular quantities that are mixed or 

 compounded, and their refpcdlive mean rates, arc given. 

 The fever.d cafes will come under the following rules. 

 I. The quantity of the ingredients, and the prices of 

 each, being given j to find the price or value of fome part 

 of the mixture. 



Rule, As the fum of the quantities given, 



Is to the fum of the produfts of each ingredient 



by its price. 

 So is any quantity of the mixture, 

 To its value. 

 Example i. A refiner, or goldfmith, hath 12 ^ of gold 

 at 4/. per J, 8 J at 4/. 5/. 3 J at 4/. 6s. Sd. and y f at 

 4/. 13^. 4//. per J ; what is an ounce worth of all thele 

 melted together ? 



By the fame rule the value of any other quantity of that 

 compofitiou is to be found : as fuppofing 7 J. 



For as 32 : 137 : : 7 : 29 j I. 

 Example 2. Suppofe it were required to mix 6 gallons of 

 wine at ^s. a gallon, 8 at 6s. and 4 at 8,f. what woidd be 

 the value of the mixture per gallon ? 



6 X 5 = 30 



8 X 6 = 48 



4 X 8 = 32 



Whole compound, 18 iio fum of produfts. 



Then 18) 1 10 ( 6-;^ or 6^. is the value fought. 



II. The prices ot the feveral ingredients, and the fum 

 paid or received for the mixture being given ; to find what 

 quantity of each was bought or fold. Divide the fum paid 

 or leceived, by the fum of the particular prices ; the quo- 

 tient is the aniVer. 



III. The ingredients of a mixture being given, to augment 

 or diminilh the mixture proportionally. 



Ride. As the fum of the particular quantities of the 



compound given, 

 Is to the whole quantity propofed to be augmented or 



leiTened ; 

 So is each particular quantity in the given compoimd. 

 To the due proportion required of that fpecie, finenefs, 



&c. 

 Example. The compound in the foregoing inflance is re- 

 quired to be augmented to 48 J : that is, 1 6 is to be added 

 to 32, how much of each ingredient mnil be taken J 



Then as 32 : 16 



(■ 



3 

 9 



32 fum 



63 



4 



li 



■ 1 6 fum. 



J. 



o per 2- 



So that there mull be 18 ^ of gold at 4 



12 4 



4i 4 



i?i 4 



Sum = 48 for proof of thv operation. 

 IV. The nature, quality, 5«c. of the feveral ingredients of 



5 o 



6 8 



13 4 



ALL 



a mixture being given, to find the temperament or degree of 

 finenefs refidting from the whole. Place the fevtral quanti- 

 ties of the mixture in rows ; againll \vhich place orderly 

 their feveral qualities or finenefs : and multiply each quan- 

 tity by its own quality or degree of finenefs ; then, as the 

 fum of the quantities is to the produds, fo is unity to the 

 quality or finenefs of the mixture. 



V. The quantities of a mixture being given ; to find the 

 particular quantities of any ingredient in any part of the 

 mixture. 



Rule. As the total of the compofition, 



Is to the quantity of any fimple in that compofition, 



So is the total quantity propofed, to be proportionably 



compounded, 

 To the quantity of each fimple to be in that propofed 



quantity. 

 Example. How much of each ingredient (or price of 

 gold mentioned in the firft cafe) is in a pound, or 12 3 of 

 the 32, being the compound given ? 



As 32 : 





at 

 at 



at 

 at 



5 



6 



13 



//. 



o per |. 



o 



8 



4 



4i 

 3 



Ji 



1 2 proof. 



VI. Given the total of a mixture, with the whole value, 

 and the values of the feveral ingredients ; to find the fevcnil 

 quantities mixed, though unequally. 



This cafe admits of two varieties : firft, where the mix- 

 ture is of two fimples ; and, fccondly, when it confifls of 

 more than of two. For the firft, the rule is — Multiply tlie 

 total of the mixt\ire by theleaft value, fubtratt the produdl: 

 from the total value ; and the remainder is the firft divi- 

 dend ; then take the faid Icaft value from the greateft va- 

 lued ingredient, and the remainder is the firft divifor. The 

 quotient of this divifion (hews the quantity of the higheft- 

 priced ingredient, and the other is the complement to the 

 whole. 



Thus, ftill referring to the firft example, and alTuming the 

 two iiril terms of it : 



Gold at 4/. per ^. 

 Ditto at 4/. ^s. 



Total of the compofition =■ 20 ^. 

 X 4 

 80 

 quantity of the higheft-priced ingredient. 



Secondly, when the quantities are more than two in num- 

 ber. 



Thefe kinds of queftions, as in tliofe of alligation alter- 

 nate, admit of various aiifwers, all of them true, and are 

 called INDETERMINATE problems. They are beft done by 

 parcel, two at a time, as in the preceding operation. 



Alligation alternate is the method of finding the 

 quantities of ingredients or fimples ncccflary to form a com- 

 pound of a given rate or quality j and it is tlic convcrfe of 

 alligation medial. 



Alligation alternate (hews the diie proportion of feveral 

 ingredients ; and counterchangcs the place of fuch exceftes 

 oi- differences as arife between the mean price and the ex- 

 tremes ; afcribing that to the greater extreme '.vhich proceeds 

 from the leffcr : and contravily. 



The rules which obtain in alligation aliernnte arc as fol- 

 low : every greater extreme is to be linked with one 

 lefter. If either of the extremes be fingle, and the other 

 extremes plural, the fingle extreme is to be linked to all 

 the reft- 



If 



Total value Rz/. 



—So 



-^ by ^/.)"T(8 the 



