RULE. 



JHindred weight. The second kind con- 

 tains all such questions wherein we are 

 left to discover, from the nature and cir- 

 cumstances of the question, that a fourth 

 proportional is sought ; and consequent- 

 ly, how the slate of the proportion, or 

 comparison of the term, is to be made ; 

 which depends upon a clear understand- 

 ing- of the nature of the question and pro- 

 portion. After the given terms are duly 

 ordered, what remains to be done is to 

 find a fourth proportional. But to re- 

 move all difficulties as much as possible, 

 the whole solution is reduced to the fol- 

 lowing 1 general rule, which contains what 

 is necessary for solving- such questions, 

 wherein the state of the proportion is 

 given ; in order to which it is necessary 

 to premise these observations. 



1 In all questions that fall under the 

 following rule there is a supposition and a 

 demand : two of the given numbers con- 

 tain a supposition, upon the conditions 

 whereof a demand is made, to which the 

 other given term belongs ; and it is there- 

 fore said to raise the question ; because 

 the number sought has such a connection 

 with it as one of these in the supposition 

 has to the other. For example : if three 

 yards of cloth cost 4/. 10*. (here is the 

 supposition) what are 7 yards 3 quarters 

 worth? here is the demand or question rais- 

 ed upon 7 yards 3 quarters, and the form- 

 er supposition. 



2. In the question there will sometimes 

 be a superfluous term ; that is, a term 

 which, though it makes a circumstance 

 in the question, yet it is not concerned in 

 the proportion, because it is equally so in 

 both the supposition and demand. This 

 superfluous term is always known by be- 

 ing twice mentioned, either directly, or by 

 some word that refers to it. Example, if 

 three men spend 20/. in 10 days, how 

 much, at that rate, will they spend in 25 

 days ? Here the three men is a superflu- 

 ous term, the proportion being among 

 the other three given terms, with the 

 number sought ; so that any number of 

 men may be as well supposed as 3. 



Rule 1. The superfluous term (if there 

 is one) being cast out, state the other 

 three terms thus : of the two terms in the 

 supposition, one is like the thing sought 

 (that is, of the same kind of thing the 

 same way applied) ; set that one in the 

 second or middle place ; the other term 

 of the supposition set in the first place, 

 or on the left hand of the middle ; and 

 the term that raises the question, or with 

 which the answer is connected, set in the 

 third place, or on the right hand ; and 



thus the extremes are like one another, 

 and the middle term like the thing sought : 

 also the first and second terms contain 

 the supposition, and the third raises the 

 question; so that the third a d fourth 

 have the same dependence or c nnection 

 as the first and second. 2. Make all the 

 three terms simple numbers of the low- 

 est denominations expressed, so that the 

 extremes be of one name. Then, 3. Re- 

 peat the questions from the numbers thus 

 stated and reduced (arguing from the sup- 

 position to the demand), and observe 

 whether the number sought ought to be 

 greater or lesser than the middle term, 

 which the nature of the question, rightly 

 conceived, will determine; and, accord- 

 ingly, multiply the middle term by the 

 greater or lesser extreme, and divide the 

 product by the other, the quote is like 

 the middle term, and is the complete an- 

 swer, if there is no remainder ; but if 

 there is, then, 4. Iteduce the remainder 

 to the denomination next below that ot" 

 the middle term, and divide by the same 

 divisor, the quotient is another part ot 

 the answer in this new denomination. And 

 if there is here also a remainder, reduce 

 it to the next denomination, and then 

 divide. Go on thus to the lowest deno- 

 mination, where, if there is a remainder, 

 it must be applied fraction-wise to the 

 divisor; and thus you will have the com- 

 plete answer in a simple or mixed wum- 

 ber. 



Note. If any of the dividends is less 

 than the divisor, reduce it to the next 

 denomination, and to the next again, t'ril 

 it be greater than, or equal to, the divisor. 



EXAMPLES. 



Quest. 1. If 3 yards of cloth cost 8s. 

 what is the price of 15 yards ? Answ. 40s. 

 or 21. 



Work. 



yds. s. yds. 

 3815 

 15 



3)120(40*. 



Explanation. 3 yards and 8s. contain 

 the supposition, and 8s. is like the thing 

 sought ; therefore 8,v. is the middle term, 

 and yards on the left : then the demand 

 arises upon 15 yards, and therefore it is 

 on the right. Again, from the nature of 

 the question, it is plain that 15 yards re- 

 quire more than 3 yards, i. e. the answer 

 must be greater than the middle term ; 

 wherefore 8s. is to be multiplied by 15 

 yards; the product is 120s. which, divided 



