rical product of 3 and 4 will be equal to the number of little squares, each of whose sides is equal to the line BD, contained in the rectangle AC; for in both the number is twelve: the reason of which is this; the rectangle may be considered as produced by adding together three rectangles, each equal to AD, whose base is the line AB, which represents the multiplicand 4, and its height BD, or unity, and which consequently contains 4 little squares of unity; and the numerical product of 4 into 3 is produced by taking the number 4 three times, or (to use the same expression as before) by adding together three numbers, each of which contains 4 units. And this reasoning, and consequently the analogy between products and rectangles derived from it, evidently holds of any other numbers whatsoever. Consequently, whatever is demonstrated concerning the proportions of geometrical rectangles (those proportions being the same with the proportions of the numbers of the squares of the line that represents an unit contained in them), is true likewise of the proportions of the products of numbers proportional to the sides of those rectangles. Thus, for instance, all the second book of EUCLID'S Elements is true of products as well as rectangles. 11. But the proper proof of this principle in the zase of whole numbers, or that which is purely alge braic, or derived immediately from the nature of quantity in general, without having recourse to the properties of any particular kind of quantity, as lines or surfaces, is that which EUCLID has given of it, in the first proposition of the fifth book of his Elements; he having there demonstrated, that, if there be two sets of quantities, a, b, c, and ma, mb, mc, whereof each quantity in the latter set contains each quantity in the former set m times, the whole latter set will contain the whole former set m times; that is, when expressed in algebraic characters, ma+mb +mc is a+b+cxm. = (To be continued.) ARTICLE XXIV. NEW QUESTIONS to be solved in the next Number. I. QUEST. 73. By Jacob Winooski, Green Mountains, Vermont. Five political vagabonds A, B, C, D and E, are to be transported from the city of New-York. The transportation of A, B, and C, will cost 101; of B, C, and D, $113; of C, D, and E, $116; of A, D, and E, $112; and of A, B, and E, $116: what will be the expence of transporting each. II. QUEST. 74. By Peter Spangler, York-Town, Penn. From the vertical angle of a given right angled plane rectilineal triangle, it is required to draw a straight line to cut the base, so that the area of the oblique angled triangle intercepted between the hypotenuse and the required line, the square of the base of this oblique angled triangle, and the square of the required line, taken together, may be equal to the square of the hypotenuse. III. QUEST. 75. By James M'Ginness, Harrisburg, Pennsylvania. In page 142 of the Mathematical Correspondent, 1/3 it is said that -X s = (2√√√3—3) xs; and in page 145 of the same work, it is asserted that √2 ·× s=(2−√2)xs: I demand a demonstra2+1 tion of the truth or falsity of these two equations. IV. QUEST. 76. By John Smithis, Philadelphia. In extracting the roots of the powers of numbers, why do we make the number of digits in a period, equal to the exponent of the power? V. QUEST. 77. By Thomas Maughan, Quebec. Required an investigation of the principles of Plain Sailing in Navigation. VI. QUEST. 78. By John Capp, Harrisburg, Penn. Determine, by algebra, the number of degrees, &c. contained in an angle, whose cosine is equal to its. tangent. VII. QUEST. 79. By James M'Cormic, Carlisle, Penn.. In a trapeziod are given one of the two parallel sides, considered as a base, the two angles adjacent to the base, and the area, to determine the altitude or perpendicular height of the figure. VIII. QUEST. 80. By the Rev. T. P. Irving. My tortoise-shell snuff-box, ingeniously wrought, With iv'ry adorn'd and gold spangles, Has sides, ends and bottom, all square like its top, And equal its eight solid angles. A ringlet, diag'nally, roll'd o'er its lid, Completes an entire revolution, And marks, by a punctum, a curve, that contains The clew to this question's solution. Three inches, and more, by two ones, nine and four, Plac'd after the decimal sign, Sirs, Extended on Gunter's old iv'ry or box, The length of the curve, will define, Sirs. The curve, tho', exceeds the diag'nal within ;Take out, then, by simple subtraction, One, three nines, six, eight, nine, six, one and a five, Moreover, this box cramm'd with choice Macuba, Retaining the depth, make the length to the breadth IX. QUEST. 81. By Jacob Cochran, Rockland, New Jersey. How far must a Ship sail on a S. W. by S. course, from a port in Lat. 38° 15′ N. before her Latitude and difference of Longitude become equal? X. PRIZE QUEST. 82. By John D. Craig, Baltimore. In a circular arc less than a quadrant, it is required to find the relation between that arc and its cotangent, when the rectangle of the arc and its cosine is a maximum. ARTICLE XXV. SOLUTIONS of the QUESTIONS proposed in ARTICLE XXIV. I. QUEST. 73. Solved by the proposer, Jacob Winooski, Green Mountains, Vermont. Let the expenses of transporting A, B, C, D and E, be respectively represented by a, b, c, d, and e dollars. Then by the question, a+b+c=101 b+c+d=118 c+d+e=116 a+d+e=112 a+b+c=116 Now the sum of these five equations, divided by 3, gives the equation F, or a+b+c+d+e=186; and the five original equations, severally subtracted from Fleaves G, or d+e=85 H, or ate=73 J, or b+c=74 K, or c+d=70. Lastly it is evident that F―J+G=a+b+c+d+e— b+c+d+e=a 186-15927; whence, from the equations I, J, K and H are found b=43, c=31, d=39, and e=46 dollars, the sums required. H. QUEST. 74. Solved by the proposer, Peter Spangler, York-Town, Pennsylvania. Let ABC be the given right angled plane rectilineal triangle, and BD the required straight line; then by mensuration AD.BC=the area of the oblique angled triangle ABD. Now by the questionAD.BC+AD2+BD2= AB; and by prop. 12. book 2. Euc. A D C |