RULE III*. WHEN one of the ingredients is limited to a certain quantity: take the difference between each price, and the mean rate as before; then say, as the difference of that simple, whose quantity is given, is to the rest of the differences severally; so is the quantity given, to the several quantities required. EXAMPLES. 1. How much wine at 5s, at 5s 6d, and 6s the gallon, must be mixed with 3 gallons at 4s per gallon, so that the mixture may be worth 5s 4d per gallon? 10 20 3 6 Ans. 3 gallons at 5s, 6 at 5s 6d, and 6 at 6s. 2. A grocer would mix teas at 12s, at 10s, and at 6s per lb, with 20lb at 4s per lb: how much of each sort must he take to make the composition worth 88 per lb? Ans. 20lb at 4s, 10lb at 6s, 10lb at 10s, and 20lb at 12s. POSITION. POSITION is a rule for performing certain questions, which cannot be resolved by the common direct rules. It is sometimes called False Position, or False Supposition, because it makes a supposition of false numbers, to work with in the same manner as if they were the true ones, and by their means discovers the true numbers sought. It is sometimes called Trial-and-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors. Position is either Single or Double. suspecting the workmen had debased it by mixing it with silver or copper, he recommended the discovery of the fraud to the famous Archimedes, and desired to know the exact quantity of alloy in the crown. Archimedes, in order to detect the imposition, procured two other masses, the one of pure gold, the other of silver or copper, and each of the same weight with the former; and by putting each separately into a vessel full of water, the quantity of water expelled by them determined their specific gravities: from which, and their given weights, the exact quantities of gold and alloy in the crown may be determined. Suppose the weight of each crown to be 101b, and that the water expelled by the copper or silver was 921b, by the gold 521b, and by the compound crown 641b: what will be the quantities of gold and alloy in the crown? And the sum of these is 12+28 = 40, which should have been 10; therefore by the Rule, 40: 10 12: 3lb of 40: 10 28 71b of gold the answer. * In the very same manner questions may be resolved when several of the ingredients are limited to certain quantities, by finding first for one limit, and then for another. The last two rules can need no demonstration, as they evidently result from the first, the reason of which has been already explained. 93 SINGLE POSITION. SINGLE POSITION is that by which a question is resolved by means of one supposition only. Questions which have their result proportional to their supposition, belong to single position: such as those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself, or any parts of itself, a certain proposed number of times. The rule is as follows: TAKE or assume any number for that which is required, and perform the same operations with it, as are described or performed in the question: then say, as the result of the said operation is to the position or number assumed; so is the result in the question to a 4th term, which will be the number sought*. EXAMPLES. 1. A person after spending and of his money, has yet remaining 607; what had he at first? 2. What number is that, which being increased by,, and of itself, the sum shall be 75? Ans. 36. 3. A general, after sending out foraging and of his men, had yet remaining 1000 what number had he in command? Ans. 6000. 4. A gentleman distributed 52 pence among a number of poor people, consisting of men, women, and children; to each man he gave 6d, to each woman 4d, and to each child 2d: moreover there were twice as many women as men, and thrice as many children as women: how many were there of each? Ans. 2 men, 4 women, and 12 children. 5. One being asked his age, said, if 3 of the years I have lived be multiplied by 7, and of them be added to the product, the sum will be 219: what was his age? Ans. 45 years. DOUBLE POSITION. DOUBLE POSITION is the method of resolving certain questions by means of two suppositions of false numbers. *The reason of this rule is evident, because it is supposed that the results are proportional to the suppositions. To the double rule of position belong such questions as have their results not proportional to their positions: such are those, in which the number sought, or their parts, or their multiples, are increased or diminished by some given absolute number, which is no known part of the number sought. RULE *. TAKE or assume any two convenient numbers, and proceed with each of them separately, according to the conditions of the question, as in single position; and find how much each result is different from the result mentioned in the question, calling these differences the errors, noting also whether the results are too great or too little. Then multiply each of the said errors by the contrary supposition, namely, the first position by the second error, and the second position by the first error. Then, if the errors are like, divide the difference of the products by the difference of the errors, and the quotient will be the answer. But if the errors are unlike, divide the sum of the products by the sum of the errors, for the answer. Note. The errors are said to be like, when they are either both too great or both too little; and unlike, when one is too great and the other too little. EXAMPLE. 1. What number is that, which being multiplied by 6, the product increased by 18, and the sum divided by 9, the quotient should be 20? * Demonstr. The rule is founded on this supposition, namely, that the first error is to the second, as the difference between the true and first supposed number, is to the difference between the true and second supposed number; when that is not the case, the exact answer to the question cannot be found by this rule. The algebraist will know at once to what class of questions this property belongs, when it is stated that it is only to such as rise no higher than a simple equation. When it is by means of one equation, and one unknown, the solution can be obtained by single position: and when it involves two propositions, one of which cannot be mentally eliminated, we must have recourse to double position. When the conditions lead to a quadratic or higher equation, the condition above named, upon the hypothesis of which the rule is formed, does not take place, and the solution obtained by it, cannot therefore be more than an approximation. The degree of approximation, and some other particulars, may be seen discussed in a note under this head in the Algebra. That the rule is true, under this limitation, may be thus proved. Let a and b be the two suppositions, and A and B their results, produced by similar operations; also r and s their errors, or the differences between the results A and B from the true result N; and let a denote the number sought, answering to the true result N of the question. Then is NA=r, and NBs, or B - Ar—s. And, according to the supposition on which the rule is founded, rs:: x- ax-b; hence, by multiplying extremes and - rb = sx — sa; then by transposition, ra sa; and, by division, the number sought, which is the rule when the results are both too little. means, rx rb -sa Sierb If the results be both too great, so that A and B are both greater than N; then N — Ar B=- s, or r and s are both negative; hence - −r: −8::+r+s, therefore rsx in the former case. - s :: x — α : xb, but ax-b; and the rest will be exactly as But if one result a only be too little, and the other B too great, or one error r positive, and the other s negative, then the theorem becomes a =when the errors are unlike. rb + sa Or, by single position :—the increase given to the product of the number being 18, and this as well as the said product divided by 9, will give the question under the following form. Required a number, to six-ninths of which if 2 be added, the sum shall be 20; or again, more simply, six-ninths of which is 18. Then suppose 18 the number. Then, 8 × 18 = 12, which is too little by 6. Then 12 18:18: 32427 Answer. The advantage in practice of double position is, that instead of requiring any mental preparations similar to those above mentioned, it renders the whole process mechanical; and which are indeed tantamount to as many algebraical ones, bearing in fact a great resemblance to the unsymbolic algebra of the Arabians and Persians and Indians. RULE II. FIND, by trial, two numbers, as near the true number as convenient, and work with them as in the question; marking the errors which arise from each of them. Multiply the difference of the two numbers assumed, or found by trial, by one of the errors, and divide the product by the difference of the errors, when they are like, but by their sum when they are unlike. Or thus, by proportion: as the difference of the errors, or of the results (which is the same thing), is to the difference of the assumed numbers, so is either of the errors, to the correction of the assumed number belonging to that error. Add the quotient, or correction last found, to the number belonging to the said error, when that number is too little, but subtract it when too great, and the result will give the true quantity sought *. * For since, by the supposition, r:s:: x − a x-b, therefore by division, r—s: s:: b—a: xb, or as B- Ab-as; x b, for B - A is rs; which is the 2nd rule. Of course the remarks upon the approximation of the first rule apply likewise to the present one. EXAMPLES. 1. Thus, the foregoing example, worked by this 2nd rule, will be as follows: sum of errors 8) 24 (3 subtr. from the position 30 leaves the answer 27 ་ Or, as 22 14: 30 18, or as 8 12 2 3 the correction, as above. 2. A son asking his father how old he was, received this answer: your age is now one-third of mine; but 5 years ago, your age was only one-fourth of mine. What then are their two ages? Ans. 15 and 45. 3. A workman was hired for 20 days, at 3s per day, for every day he worked; but with this condition, that for every day he did not work, he should forfeit 1s. Now it so happened, that upon the whole he had 27 4s to receive: how many of the days did he work? Ans. 16. 4. A and B began to play together with equal sums of money: A first won 20 guineas, but afterwards lost back of what he then had; after which B had four times as much as A: what sum did each begin with? Ans. 100 guineas. 5. Two persons, A and B, have both the same income. A saves of his; but B, by spending 507 per annum more than A, at the end of 4 years finds himself 100% in debt what does each receive and spend per annum? Ans., they receive 1257 per annum; also A spends 1007, and B spends 150% per annum. PRACTICAL QUESTIONS IN ARITHMETIC. 4808 QUEST. 1. The swiftest velocity of a cannon-ball is about 2000 feet in a second of time. Then in what time, at that rate, would such a ball move from the earth to the sun, admitting the distance to be 100 millions of miles, and the year to contain 365 days 6 hours? Ans. 8131 years. QUEST. 2. What is the ratio of the velocity of light to that of a cannon-ball, which issues from the gun with a velocity of 1500 feet per second; light passing from the sun to the earth in 8 minutes? Ans. the ratio of 704000 to 1. QUEST. 3. The slow or parade-step being 70 paces per minute, at 28 inches each pace, it is required to determine at what rate per hour that movement is? Ans. 1 miles. QUEST. 4. The quick-time or step, in marching, being 2 paces per second, or 120 per minute, at 28 inches each; at what rate per hour does a troop march on a route, and how long will they be in arriving at a garrison 20 miles distant, allowing a halt of one hour by the way to refresh ? the rate is 3 miles an hour. Ans. {and the time 74 hr, or 7h 174 min. QUEST. 5. A wall was to be built 700 yards long in 29 days. Now, after 12 men had been employed on it for 11 days, it was found that they had com |