by-cy). It is thus evident that, since there is a negative term c in the multiplicand, the subtrahend or diminishing product must itself be diminished by cy. But since anything which decreases the subtrahend increases the remainder, this term cy will appear in the final product with a positive sign; thus (ax+bx-cx)—(ay+by—cy) 3. It will be observed that in "Zeno's Para- multiplicand into the negative term, viz. ay+by dox," the length of time, in which Achilles cancy, i. e. (a+b—c) (x—y)=(ax+bx—cx)—(ay+ run a mile is not stated. Whatever this may be, if we assume it as the unit of time, we shall have an infinite series 1. .01 .0001 .000001 &c., representing the successive periods of time, which Achilles will occupy in reaching the successive positions of the tortoise. Now, although this series will consist of an infinite number of terms, as is shown in the statement of the Paradox, yet |=ax+bx-cx-ay-by+cy. its sum will be finite, since it is a decreasing geometrical series, of which the first term is 1, and the ratio .01. If, now, we divide the first term by the difference between the ratio and unity, we shall obtain as the sum of the series 1 1-99, which therefore expresses the length of time occupied by Achilles in overtaking the tortoise. ADDITIONAL QUESTION. Does the fluid in a tunnel always whirl in the same direction? If so, is the direction towards the right or the left? What determines it? For the Schoolmaster. D. G. The sophistry becomes apparent, when an infinite number of periods of time is distinguished Solution of Question 6, in the October No. from an infinite period. NOTE. It might seem that a similar course of reasoning would show the solution of the second question to be fallacious. It will be noticed however, that in that, although the number of terms in the series representing the successive periods of time is determined by the number of terms in the infinite decreasing series representing quantity of matter, yet the terms themselves are equal periods of time, of one year each, whose sum must be infinite since their number is infinite. The apparent paradox in question 6, arises from the common, though erroneous, habit of considering a quantity affected with the minus sign as having an actual existence. A minus quantity has no absolute existence; it depends upon some unknown quantity, which though not expressed, must be mentally connected with it. The sign, minus, is simply a sign of subtraction; it denotes that the quantity which it precedes is to be taken from some other, either expressed or understood. E. G.a. denotes that a is to be subtracted from some unknown quantity, which we may represent by x, so that—a is really x-a. 4. The sophistry in this Paradox is almost too apparent to require a statement. Its absurdity was illustrated in a most practical manner by the surgeon, for whom Diodorus sent, when he had It is well known that the product of a by dislocated his shoulder. He proved to the Sophist, by a sort of argumentum ad hominem, that is +ab. This has been explained in the following manner. Let -a be regarded as a subon his own principles the humerus could not be removed from the socket. We may well believe tracted from some unknown quantity, say z, and that in this case Diodorus was forced to acknowl-as b subtracted from some other unknown edge that a thing might sometimes be moved quantity, as y. from a place where it ought to be, into a place where it ought not to be. 5. On account of the oblate form of the earth, the source of the Mississippi, being in a much higher latitude than the mouth, and not being among lofty mountains is absolutely nearer the centre of the earth than the mouth. Then instead of a and b for multiplicand and multiplier, we have z—ɑ and y-b. Now multiply x-a by y-b and the result is xy-ay-bx+ab. Let us examine the process in detail. x multiplied by y gives ry, but the whole of x ought not to be multiplied by y; the multiplicand is x diminished by a, therefore the result is as much too large as the product of a by y, or ay. The product of x-a by y is then xy-ay. The result would be correct if x-a were to be The centrifugal force of the earth, however, is sufficient to make the river flow towards the low-multiplied by the whole of y, which it is not, but er latitudes, where the rapidity of revolution is greater. Were the earth gradually to cease revolving, the Mississippi would undoubtedly commence running towards the north. 6. Suppose a+b-c is to be multiplied by x-y. It is evident that the product of the multiplicand into the positive term of the mutiplier, viz. ax+ bx-cx is to be diminished by the product of the by y diminished by b. The product xy-ay, then, is too large in the same proportion that y is greater than b. We therefore multiply a by b=bx, and subtract it from the product xy-ay changing it into xy-ay-bx. But the whole of ought not to be multiplied by b, for x-y is the multiplicand; so that the result xy-ay-bæ, is actually too small, as much too small as the pro duct of b by a, or ab. Add, then, ab to xy-ay br, and we obtain xy-ay-bx+ab, which is the true result. Now when two minus quantities, such as a and b, are given to be multiplied together, we must mentally supply the and y, and we shall have no difficulty in understanding why -a multiplied by should produce +ab. F. H. H. For the Schoolmaster. PROVIDENCE, Dec. 5. 1857. MR. EDITOR-I send the following solutions to the problems in the October number of the Schoolmaster, hoping they may be found worthy of a place in the next number. 1. The rods must be placed touching at their ends, and at right angles to each other. 2. The wheel will never cease to revolve. 3. The time required for Achilles to overtake the tortoise is equal to the sum of the series, 1, .01, .0001 .000,001, &c., carried to infinity, the unit of the series being the time required for Achilles to pass over the first mile. The argument amounts to the following: Since the number of terms in this series is infinite, the time must be infinite, or, it amounts to the same thing as saying that the sum of an infinite decreasing series is infinity; which is obviously wrong. 4. The fallacy is in the hypothesis. Motion is a change of place. If anything is moved, it must be taken from one place and put in another place. The hypothesis is equivalent to the following: If a body is taken from one place and put in another place, it is either taken from a place where it is and put in another place it is, or, it is taken from a place where it is not, and put in a place where it is not. Thus stated, the fallacy is evident, for a body cannot be in two places at the same time. 5. The water is forced up to the mouth of the river by the centrifugal force generated by the earth's motion around its axis. 6. a×+5—5a; —ax+1=—a, which is greater than -5a; -ax0=0, which is greater than -a. Hence we see that the smaller the multiplier of ■ negat ve quantity, the larger is the product. Now, since the product of -a multiplied by zero equals zero, the product of a multiplied by a quantity less than zero, or, by a negative quantity, must be greater than zero, that is, it must be a positive quantity. D. S. MATHEMATICAL PUZZLE.-Given a rectangular quadrilateral, whose adjacent sides are in the proportion of 16:9. It is required to cut this into two pieces, such that they can be placed together in the form of a square. A gentleman has two hundred rods of wall to build. He engages two men to build the wall. The stones, with which the wall is to be built, must be drawn farther to build one end of the wall than the other. He therefore engages one man to build for seven-eighths of a dollar per rod, and the other for nine-eighths of a dollar per rod. The two men are to build the two hundred rods. Each man is to build one hundred dollars worth. How many rods will each build, to answer all the conditions of the question? Brother Mathematicus, when you have become satisfied that you can't solve the above, and have found an absurdity in it, please tell us wherein lies the absurdity. EXAMPLE IN ANALYSIS.-If 4 men can saw.15 cords of oak in the same time that 5 men saw 14 cords of hickory, and if 3 men saw 18 cords of hickory in 3 days by working 9 hours a day, how many hours a day must 7 men work, to saw 84 cords of oak in 6 days? OUR BOOK TABLE. THE NATIONAL FOURTH READER; containing a course of instruction in Elocution; exercises in Reading and Declamation; and copious Notes, giving the pronunciation and definitions of words, biographical sketches of persons whose names occur in the reading lessons, and the explanation of classical and historical allusions. By Richard G. Parker, A. M.. and J. Madison Watson. A. S. Barnes & Co., New York. This is the Fourth Book in the National Series of Readers, published by the above named enterprising house. The course of instruction in Elocution" is quite extended, occupying 44 pages. The "exercises in Articulation," are admirably adapted to promote accuracy of pronunciation, so important to good reading. We quote a single sentence for illustration: "The rival robbers rode round and round the rough and rugged rocks that rear their hoary heads high in the air." The foot notes on each page are very valuable. Upon the selections for reading, we are not competcnt to express an opinion without a trial in the school-room. THE ATLANTIC MONTHLY for December, is, in our opinion, more readable than the first number. This work is destined to take a very high rank among our monthlies. We quote the following from the "Correspondent of the Evening Post." "The arrangements with foreign and home writers of eminence were consummated by Mr. F. H. Underwood, reader of the publishing firm, and they are of so permanent and binding a character on both sides as to insure constancy and regularity on the part of the contributors whose names are expected to give the Magazine popularity. In this way it is hoped to obviate the chief cause of the decline of former first-class periodicals, viz: the dropping off of eminent writers, whose efforts to prove effective to the magazine should not be occasional, but apparent in every number." GREENLEAF'S NEW PRIMARY ARITRMETIC. GREENLEAF'S INTELLECTUAL ARITHMETIC.Robert S. Davis & Co., Boston. The first named of these books is a simple treatise, designed for the primary school. The " "Intellectual" is an advanced work, very much after the model of Warren Colburn's world-renowned book, giving a fund of examples and exercises, which will probably not be exhausted in every school, to say the least. 3. That the Dictionary cannot serve as such, because it does not classify words according to prevailing mispronunciations, and because it contains, besides, a great deal of extraneous matter, (extraneous, that is, to this particular purpose,) and thereby perplexes and discourages the pupil. And 4. That, on such a subject, it is no objection to a text-book that it does not please every body, since that must necessarily be the case with any and every text-book that could be compiled. Taking these propositions for granted, it claims to have met an existing want in a plain. practical manner. All it asks is, a fair trial, and to stand or fall on its own merits." We think, if it has "a fair trfal," it will stand and not fall. It will be a valuable book for teachers and for drill exercises in classes. We hope it will be extensively used. THE importance of the thorough etymolog- tual love was strengthened by common woes. ical study of English has been of late earnest-"Friend" was originally a present participle ly commended to many of our teachers. It of the verb "frian," "to love." Affection is urged that such study discovers unsuspect-is regarded in this derivation as essential to ed richness of meaning in many familiar words, true friendship. The termination "ing" has reveals to us the wondrous power of our lan- taken the place of this old participial form in guage, and gives greater skill in the use of it." nd"; though Spenser (F. Q. B. 1. Canto VII. The following analysis of a well known pas- 29.) has “glitterand” for "glittering. sage is offered as an illustration of this kind Supply the indefinite "some one' ་་ friand," of study: i. e. "loving," and you have a friend. The word “Friends, Romans, countrymen, lend me your which was once peculiar to one of the Saxon has the common English plural termination ears; I come to bury Cæsar, not to praise him. JULIUS CÆSAR, Act III., Scene 2. declensions. To this conciliatory word Shakspeare makes Antony add one which to Antony's fellow citizens was full of honor. To be a Roman was to be noble. "With a great sum," said the chief captain to Paul, "obtained I this freedom." "But I," said the apostle with becoming self-respect, "was free born." This form of address was therefore flattering. Our English word "Roman comes to us through the Norman French-itself a descendant of Few, if any, of these words are invested with any peculiar historical interest. The meaning of none of them has undergone any marked change in the lapse of time. They the Latin, and therefore called a Romance have neither been exalted nor degraded from their first estate. Most of them are short; most of them are Saxon in their origin. language from the city where that ancient tongue was the vernacular speech. The third word of salutation was not less winning, With a multitude of other words expres-"countrymen," partakers with me of nationsive of social relations and their fitting emo- al privileges, honors, joys, and reverses. The tions, or associated with home and the fire-speaker puts himself on the same footing with side, the Anglo Saxon tongue has given us the his hearers. This word "countrymen" is first word of this oration. Side by side with one of those hybrid words common in our father," "mother," "brother," and "sis- tongue, in which one element is contributed by one language, another element by another arily, to put into a grave. Its derivative noun language. Thus the Norman French gives us in Saxon is "buhr" or "byhr," which will "country;" the Anglo Saxon gives us "man;" remind German scholars of "Burg," "a castle,” the two together implying community of na- or "stronghold," and will remind us all of comtional origin and abode. In Shakspeare's mon terminations for the names of places, time, as now among those who use English Edinburgh, Dryburgh, Shrewsbury, Roxbury. correctly, it was not thought necessary to pre- The orator comes to put in a place of safety fix to the word the epithet "fellow" to ex- these mangled remains of Cæsar. But whence press this social idea. It was left for those originates the preposition "to" between these who take for one of their maxims, "Our verbs? The Saxon primitive "byrgan" has country right or wrong to commit this bar- no such prefix, nor do the classical tongues barism, by speaking of their " fellow country-exhibit such a construction in the simple in men." finitive. Saxon etymology offers the key to The word "country," the first element in this complex word is from the French. In Chaucer's Prologue to the Canterbury Tales, written before the English had assumed its present forms the foreign spelling is retained, "contree." This Romance word has no immediate Latin ancestor. Its nearest Roman relative is the military term "conterraneus," an auxiliary and a principal verb as e. g. "I applied to men of the same land. shall go," "I will speak," are illustrations of this construction. The dative of the infinitive—a strict gerundive form like the Latin "Lend me your ears," not give, but retain your control over your hearing and your as"Lend" and "loan" related words," Amandum "-ended in "enne in Saxon, sent. as in Polonius' address in Hamlet, "Neither a borrower, nor a lender be, For loan oft loses both itself and friend,” have this idea in common, the perpetuation of the right of the original owner of the thing lent. and was preceded by the preposition "to." The termination has been dropped, but the preposition is retained, and standing before the verb governs it as though it were a noun. It need scarcely be remarked in this connexion that the practice of inserting a qualifying adverb between the preposition and the verb It may be remarked once for all, in respect is forbidden by this theory of their mutual to the several personal and possessive pro-relation. Cæsar is of course directly from nouns of this passage, "we," "you," "your," the Latin. "Not," "a surly sort of word," "it," and "their," that they come with very as Horne Tooke calls it, is contracted from slight verbal change from that vigorous tongue "naught," itself a compound of "no" and which, contributing five eighths of the words aught," "awhit," "anything." in our language, gives us all those frequently occurring words which, expressing relation, furnish the essential elements of all our sentences. The Latin, the French, the German, and the Anglo Saxon, denote the organ of hearing by kindred words. "Auris," "orcille," "ohr," "ear," are so nearly alike as to suggest community of origin. " 44 "I come" is Saxon; and so also is "to One of the most frequently recurring bury." The latter expression, however, de- words in our language is the definite article serves some attention, first, for its derivation the," here prefixed to "evil." In common and meaning, and secondly, for its form. Its with its relative "that," it has received a cuSaxon form is "byrgan," signifying prima-rious derivation, which, though seemingly rily to put in safety, to conceal, and second-fanciful, may be received until supplanted by |