differently for the particular numbers given in the question, or for any other numbers whatever. It is agreed to represent known quantities, or those which are suppose to be given in a question, by the first letters of the alphabet, as a, b, c. Representing by a the number to be divided, and by b the given excess, the question, art. 1, may be presented generally, thus; To divide a number a into two such parts, that the greater may exceed the less by h. To resolve the question, thus stated, we denote still the less part by x; the greater will then be x + b, and we have 19. The above expression for x is called a formula for x, since it indicates the operations to be performed upon the numbers represented by a and b in order to obtain x. The translation of a formula into common language is called a rule. Thus we have the following rule, by which to obtain the less of the parts required according to the question proposed, viz. From half the number to be divided, subtract half the given excess, the remainder will be the answer. Knowing the less part, we obtain the greater by adding to the less the given excess. We may, however, easily obtain a rule for calculating the greater part without the aid of the less. Indeed since the less part is equal to 2 2' a b , if we add b to this, we have equal to the greater. But this expression may, it is easy to see, be re a b duced to whence we have the following rule, by which to find the greater part, viz. To half the number to be divided, add half the given excess, the result will be the answer. To apply these rules, let it be required to divide $1753 between two men in such a manner, that the first may have $325 more than the second. 20. The sixth question, art. 6, may be presented in a general manner, thus; To divide a number a into three such parts, that the excess of the mean above the least may be b, and the excess of the greatest above the mean may be c. Let x the least part; and x+b+c= the greatest, 3x=a- 2b-c therefore or transposing and reducing Translating the above formula into common language, we have the following rule, by which to find the least part, viz. From the number to be divided, subtract twice the excess of the mean part above the least, and also the excess of the greatest above the mean and take a third of the remainder. To obtain a formula for the mean part, we add b, the excess of the mean above the least, to the above expression for the least part, which gives for the mean In like manner the following formula will readily be obtained for the greatest part, viz. a+b+2c Translating these formulas into common language we obtain rules also for the mean and for the greatest part. 1. To apply these rules let it be required to divide $973 among three men, so that the second shall have $69 more than the first, and the third $43 more than the second. 2. A father, who has three sons, leaves them his property amounting to $15730. The will specifies, that the second shall have $2320 more than the third, and that the eldest shall have $3575 more than the second. What is the share of each ? 21. The operations necessary for the solution of this last question are, it is easy to see, the same with those for the preceding. It may therefore be solved by the same formulas. In like manner the seventh, eighth, and ninth questions, art. 6, may be solved by the same formulas. This circumstance is worthy attention, since we are thus enabled to comprehend in one the solution of a multitude of questions differing from each other not only in the particular numbers, which are given, but also in the language, in which they are expressed. Let now the following questions be generalized. 1. The sum of $3753 is to be divided among 4 men, in such a manner, that the second will have $159 more than the first, the third $275 more than the second, and the fourth $389 more than the third. What is the share of each? 2. Three men share a certain sum in the following manner; the sum of A's and B's shares is $123, that of A's and C's $110, and that of B's and C's $83. and the share of each ? What is the whole sum 123 Let x the whole sum, then x C's share, &c. 22. The seventh question, art. 15, may be stated generally, thus. A cistern is supplied by two pipes; the first will fill it in a hours, the second in b hours. filled if both run together? Let x In what time will the cistern be the time; the capacity of the cistern being sup posed equal to unity, we have Here it will be observed, that x is taken a times and also b times; whence on the whole it is taken a + b times; a+b is then the coefficient of x, and the above equation may be written, thus, Translating this formula, we have the following rule for every case of the proposed question, viz. Divide the product of the numbers, which denote the times employed by each pipe in filling the cistern, by the sum of these numbers; the quotient will be the time required by both the pipes running together to fill the cistern. EXAMPLE. Suppose one pipe will fill the cistern in 5 hours, and the other in 9 hours; in what time will it be filled if both run together? 23. The fifth question, art. 6, may be thus generalized. A gentleman meeting four poor persons distributed a shillings among them; to the second he gave b times, to the third c times, and to the fourth d times as much as to the first. What did he give to each? Let x represent what he gave to the first, we then have x + bx + cx + dx = a Let next the following questions be generalized. 1. A bankrupt wishing to distribute his remaining property among his creditors finds, that in order to pay them $175 apiece, he should want $30, but if he pays them $168 apiece he will have $40 left. How many creditors had he? 2. It is required to divide the number 91 into two such parts, that the greater being divided by their difference the quotient may be 7. 3. A person has two sorts of wine, one worth 20 shillings a gallon, and the other 12; from which he would make a mixture of 30 gallons to be worth 14 shillings a gallon. How much of each must he take. 4. Divide the number 138 into two such parts, that 5 times the first part diminished by 4 times the second will be equal to 85. 5. Three men, A, B, and C, engage in trade and gain $500, of which C is to have twice as much as B, and B $50 less: than 4 times as much as A. How much will each receive ? 6. A trader having gained $3450 by his business, and lost $2375 by bad debts, found, that of what he had left equalled the capital with which he commenced trade. What was his capital? 7. In a certain school of the pupils learn navigation, learn geometry, learn algebra, and the rest, 10 in number, learn arithmetic. How many pupils are there in all? 24. The twentieth question, art. 15, may be presented in a general manner, thus. A laborer was hired for a certain number a of days; for each day that he wrought he was to receive b shillings, but for each day that he was idle, he was to forfeit c shillings. At the end of the time he received d shillings. How many days did he work, and how many was he idle? Putting the number of days, in which he wrought, and resolving the question, we obtain d+ a c b+c EXAMPLE. A laborer was hired for 75 days; for each day that he wrought he was to receive $3, but for each day that he was idle, he was to forfeit $7. At the end of the time he received $125. To determine by the above formula the number of days, in which the laborer wrought. The three following questions may also be solved by the same formula. Why is this the case? 1. A man agreed to carry 20 earthen vessels to a certain place on this condition; that for every one delivered safe he should receive 11 cents, and for every one he broke, he should forfeit 13 cents; he received 124 cents. How many did he break? 2. A fisherman to encourage his son promises him 9 cents for each throw of the net in which he should take any fish, but the son, on the other hand, is to forfeit 5 cents for each unsuccessful throw. After 37 throws the son receives from the father 235 cents. What was the number of successful throws of the net? 3. A man at a party at cards betted three shillings to two |