ARTICLE XVI. NEW QUESTIONS to be solved in the next Number. I. QUEST. 40. By G. Baron, New-York. Without any regard to the derivation of words, I wish to know what our mathematicians understand by a fraction; and whether the terms of a fraction ought always to be considered as homogeneous quan tities. II. QUEST. 41. By R. Patterson, jun. Philadelphia. Given any two sides of a right-angled plane triangle, to discover a general rule, for finding the third side, by a numerical operation that shall be the shortest possible. III. QUEST. 42. By T. Bulmer, Sunderland, England. Prove that any number divided by 9, and the sum of its digits divided by 9, leave equal remainders; and determine whether this is an absolute property of the number 9, or merely an effect of our present notation. IV. QUEST. 43. By J. Craggs, near Richmond, Virg. VI. QUEST. 45. By Eben. R. White, Danbury, Connect. Given one side of a trapezium and the two adjacent angles to find the length of two equal sides, contiguous to the given side, when the area of the trapezium is a given quantity. VII. QUEST. 46. By W. Lenhart, York-Town, Penn. In a plane rectilineal triangle, if from one of the angles adjacent to the base a straight line be drawn perpendicular to the opposite side, then shall the rectangle contained by the side on which the perpendicular falls, and the part intercepted between the vertical angle and the perpendicular, be equal to the difference of the squares of the straight line drawn from the vertical angle to the middle of the base and the semi-base; required a demonstration. VIII. QUEST. 47. By James North, Philadelphia. A hole was made in a vessel filled with wine, and, at the same instant, a supply of water was contrived to keep the vessel constantly full: the content of this vessel was 100 gallons, and the hole discharged one gallon of the mixture every 10 seconds. Now supposing the liquor to be always equally diffused throughout the vessel, how much wine remained in the same one hour after the hole was made? IX. QUEST. 48. By R. Patterson, jun. Philadelphia. The plantations of two neighbours are separated by a zig-zag fence, and it is required to determine geometrically the position of a straight-lined fence, that shall be the shortest possible, to separate these plantations, without altering their respective areas. X. QUEST. 49. By Ben. Cheetham, Billiard-Hall. Required a general theorem for finding the centre of gravity of the frustum of any pyramidical body. XI. QUEST. 50. By the Editors. Given x+y+z=a, xy+xz+yz=b and xyz=c, to find x, y, and z. XII. PRIZE QUEST. 51. By Thomas Maughan, Quebec. [The author of the best solution to this question shall receive a handsome silver medal, value six dollars.] A ship from a port in lat. 80° N. and long. 60° W. was steered eastwardly in such a manner that her distance sailed was always equal to her difference of longitude; the earth being spherical, in what latitude would her difference of longitude be 180°, and in what longitude would her latitude be 40° N.? ARTICLE XVII. SOLUTIONS of the QUESTIONS proposed in ARTICLE XVI. I. QUEST. 40. Solved by G. Baron, New-York. The question is, "What our mathematicians understand by a fraction, and whether the terms of a fraction ought always to be considered as homogene. ous quantities?" Were we to examine the various modern treatises written on the subject, we might, probably, conclude that our mathematicians have no determinate idea of a fraction. Some have defined a fraction to be a part of a number;* but the fallacy of this definition may be readily shown: for a number is a determinate multitude, or a definite plurality of units, considered without any regard to extension, and that which is not extended cannot be parted or divided; consequently a unit is the smallest possible part of any number. But it is well known to mathematicians, that whatever a fraction may be, it is fre quently less than a unit; and hence the definition is false. It may here be observed, that a number has an infinite variety of multiples, but its parts are finite, and often exceedingly limited: thus in the number 12, its multiples are 24, 36, 48, 60, 72, &c. without end; but its parts are limited to 1, 2, 3, 4, and 6, whose denominators are 12, 6, 4, 3, and 2. A prime number has no part but a unit, and the number itself denominates the part. It is, therefore, abundantly evident that the limited nature of the parts of numbers is utterly inconsistent with the géneral sense in which fractions are universally used. Others have defined a fraction to be a part of something considered Definitions of the words multiple, part, and denominator, may be seen in page 1 and 2 of this work. as a whole. Something considered as a whole is a vague expression, to which we cannot easily annex a mathematical idea; but since, from the general usage of fractions, this something is capable of being infinitely divided into parts, it must possess extension, a property which belongs to the subject of geometry, or that of the science of magnitude; for we have already seen, that the division into parts here supposed cannot be universally applied in dividing or se parating numbers into parts or equal parcels. Again, from the general usage of fractions we know that this something, considered as a whole, and its various parts, are always expressed by means of numbers; but a magnitude and its various parts cannot be universally expressed in numbers, but by means of some known measure, considered as an infinitely divisible unit; and it is well known to mathematicians, that magnitude so expressed is called quantity. Hence then this something, considered as a whole, must be quantity. From these cursory cognitions it appears to me that quantity, and not number, is the subject of fractions, and that a fraction is a part of a quantity. Let now m represent a quantity of any kind, and n any abstract number, then, according to a general custom, will represent the nth part of my and is, therefore, a fraction. The pure number n, and the quantity m, are the terms of the fraction, and of consequence can never be homogeneous. I am well aware that, to superficial mathematicians, the conclusion here drawn will appear to contradict the scholium in page 95 of this work; but if such men ever discover what is really meant by dividing one fraction by another, the apparent contradiction will instantly vanish. m n H. QUEST. 41. Solved by the Rev. Thomas P. Irving, Newbern, North Carolina. Put h the hypotenuse, p=the perpendicular, and b➡the base of a right angled plane triangle; then from prop. 47, book 1, Euc. we deduce the three following theorems: 2 h=√p2+b2=p√✅/1+b2÷p2=b√✅/p2÷62+1. Here it is plain that the values of p and b niay be readily found by logarithms; and our friend, Mr. White, has, in page 70 and 71 Math. Corr. shown us an easy logarithmic process for finding the value of h. Hence then it appears from these theorems, that the required side may be found, 1st. by extracting the square root of the sum or difference of the squares of the other two sides; 2d. by a table of squares and square roots; 3d. by a table of difference of latitude and departure; and, 4th. by a table of logarithms. No one of these methods will, however, resolve the question by means of an operation, in all cases, the shortest possible; nor does it appear to me that the nature of the subject will afford the required theorem. The same solved by the Proposer, R. Patterson, jun. Philadelphia. Put h➡the hypotenuse, p=the perpendicular, and b=the base of any right angled plane triangle. Assume xh-p, or h-b, according as p is greater or less than b; then, from prop. 47, book 1, Euclid, we have the three following equations: 2 1. h2=h—x2+b2 or h―x2+p2.. |