61. By J. B. MOTT, NEOSHO, MO.- What will be the value of each letter of the alphabet if the product of all but a is = 1, all but b is == all but c is = 3, all but d is is= 4, and so on, to all but z is = 26. 2, 62. BY DR. H. EGGERS, MILWAUKEE, WIS.-Given four lines in a plane: to inscribe a parallelogram within them with given direction of sides. 63. BY E. P. NORTON, ALLEN, MICH.-Coasting along shore, I struck upon a shoal, and wanting to ascertain its situation exactly, I took angles with my sextant, subtended by three objects on shore, as A, B and C, whose relative positions were as follows; the distance from A to B was 10 miles, from B to C 6 miles, and the angle ABC 150°; now the angle, measured at the ship, between A and B, was 24°, and between B and C 16°. Required the distance of D, the ship's place, from each object by geometrical construction, and, calculation. 64. BY DAVID TROWBRIDGE, A. M., WATERBURGH, N. Y.-Find x the maximum value of (2) without the aid of the Calculus. 65. BY EDWARD S. FARROW, WEST POINT, N. Y.-The corner of a page is turned down, and in every position the area of the triangle is 2 square inchs; find the locus of the angular point. 66. BY ARTEMAS MARTIN, ERIE, PA.-A speaks the truth 6 times out of a; B, d times out of c; and C, n times out of m. C says that B told him that A said a certain event transpired. Required the probability that the event occurred. 67. By G. W. HILL.-Mt. Shasta in California, in form, is approximately a right cone whose altitude is 2 and the radius of its base 5 miles, and is composed of homogenious rock of density 2.75. What is the angular deflection of the plumb-line at the base of the mountain, the earth being supposed a sphere without rotation, 3956 miles in radius and of mean density 5.67. 68 BY W. C. CALDWELL. [From Notes and Queries, No. 2, by request.] Let the radius vector of a spiral make a revolution and a half; then pass a curved surface through the locus of the spiral perpendicular to its plane; then place a light at the pole. Required the equation of the spiral when the rays of light are most focalized at the mouth of the spiral. Erratum. On page 36, line 3, for "CN" read CL. EDITORIAL NOTES. Since the publication of the Jan. No. of the ANALYST we have received the first two Nos. of "Educational Notes and Queries," a neat 16 page Monthly, edited and published by Prof. Wm. D. Henkle, Salem, Ohio. Of this publication we need only say that, from Prof. H's. extensive experience as Teacher, Superintendent of Public Schools and State Superintendent, and from our personal knowledge of his line of thought and study, we think he is peculiarly qualified to conduct successfully such a publication, and feel sure that Educational Notes and Queries will be found to be not only interesting and useful to teachers, but will be a valuable assistant both in science and literature to all students of every grade both old and young.-Terms: $1.00 a year in advance. We have likewise received the Feb. No. of "The Michigan Teacher," edited and published by H. A. Ford, Niles, Michigan.-Terms: $1.50 per year. We do not propose to review School Journals, whatever may be their excellencies or defects. We embrace this opportunity, however, of endorsing, most emphatically, a very able article on Compulsory School Attendance which appears in this No. of the Teacher. In this article Mr. Ford has shown that compulsory school attendance is not only not in accordance with the spirit of our government, but, as an agency to effect the required object, is nugatory and even prejudicial. Also, "Text-books on Geometry": An essay read before the Educational Institute of South Carolina, by Prof. C. H, Judson of Furman University, Greenville, S. C. In this essay Prof. Judson has given a very full and clear definition of the term limit, as used in mathematics, which will assist the student, we think, in understanding the exact signification of the term. He has also pointed out various inaccuracies that occur in many of the text-books on geometry used in our schools and colleges. Most of the objections specified we think are valid, and we commend the essay to students in mathematics. We are sorry to learn that a number of our subscribers for Vol. I, have concluded to discontinue their subscriptions. The reason assigned is that most of the subjects discussed are too difficult; and some of our friends advise us to make the journal more elementary, and, for the purpose of increasing its circulation, to give small premiums for clubs, and prizes for best solutions, &c. We are thankful for the advice, but have to reply that the experiment of our publication was inaugurated, not with the hope of being able to make it popular by presents and bribes, but for the purpose of affording a medium for the community and interchange of thought by students and teachers of mathematics. Hence we do not expect, nor do we desire, that any person should buy our publication unless he believes it to be worth to him as much as it costs; or unless he desires to promote the study of mathematics and believes that in supporting the ANALYST he will contribute to that end; and we are pleased to state that several of our subscribers, with that end in view, as we believe, have generously contributed to the support of our publication, for extra copies, sums varying from $10 to $25. In reply to our friends' advice we say, therefore, that the ANALYST will continue to be a medium for the interchange of thought by its subscribers, and that none of them will be excluded from a reasonable share of its space, proportional, however, to what the editor may believe to be the importance of the communication in the estimation of the majority of his readers. This No. is sent to all the subscribers for Vol. I who have not notified us that they wished to discontinue their subscription. We request that all who receive this No. and who have not paid for, nor communicated their wishes in relation to, Vol. II, will at once inform us whether they wish to be continued as subscribers, or remit the amount; as we will limit the edition of the next No. to the number of paying subscribers. Those who have received the 1st and 2nd Nos. of Vol. II, and who do not wish to be continued as subscribers, will please remit, for the two Nos., 70 cts. THE ANALYST. VOL. II. MAY, 1875. PERFECT CUBES. No. 3 BY PROF. W. D. HENKLE, SALEM, OHIO. In my first article on this subject (See ANALYST Vol. I. No. 5.) I explained how the tens' figure of the root of a perfect cube can be obtained from the tens' figure of the cube. In this article I shall extend the discussion to the hundreds' figure of the root. In order not to use too much space in the ANALYST I adopt a brief mode of indicating the facts, which may be readily comprehended by a reference to the former article. The figures after the braces are the endings of the perfect cubes and the subs the endings of the roots. The figures in the upper rows before the braces represent the hundreds of the cube and those in the lower rows the corresponding hundreds of the root. The direction after each brace refers to the mode of obtaining any figure in the lower from the corresponding figure in the upper row. (I use for brevity the word figure instead of number represented by the figure). In each case of multiplication or of multiplication and addition the tens are to be cast out or the left-hand digit is to be rejected from the prod uct or sum. 1 2 3 4 5 6 7 8 9 0 1 7 4 1 8 5 2 9 6 30j 1 2 3 4 5 6 7 8 9 0 01 01 1171 ; multiply by 7 and cast out the tens. 4 1 8 5 2 9 6 3 0 72141; mult. by 7, add 7, and cast out the tens. 1 2 3 4 5 6 7 8 9 0 1 6 3 0 7 4 1 8 5 29 1 2 3 4 5 6 7 8 9 0 816 31 61 mult. by 7, add 9, and cast out the tens. 9 6 3 0 7 4 1 8 5 24181; mult. by 7, add 2, and cast out the tens. 52 1 2 3 4 5 6 7 8 9 0 5 2 9 6 3 0 7 4 18 1 2 3 4 5 6 7 8 9 0 3 0 7 4 1 8 5 296 51 51; mult. by 7, add 8, and cast out the tens. 6121; mult. by 7, add 6, and cast out the tens. 5 1 2 3 4 8 1 4 7 1 1 3 4 6 9 2 5 1 2 3 4 9 2 5 8 7 4 1 8 5 6 7 8 9 0r 3 0 7 4 15 7191; by 7, add 5, and cast out the tens. 9131; X by 7, add 1, and cast out the tens. 6 7 8 9 010387; X by 3, add 5, and cast out the tens. 0 3 6 9 253377 01 5 6 7 8 9 0 1 8 1 4 7 0 31317; X by 3, add 3, and cast out the tens. 5 6 7 8 9 0 1 4 7 0 3 62347; X by 3, add 6, and cast out the tens. 36 1 2 3 4 5 6 7 8 9 0 3 6 9 2 5 8 1 470 1 2 3 4 5 6 7 8 9 01 5337 7397; X by 3, add 2, and cast out the tens. 8327 6367; by 3, add 9, and cast out the tens. 9357; X by 3, add 7, and cast out the tens. 0743; X by 3, add 5, and cast out the tens. 1773 2703; X by 3, add 0, and cast out the tens. 4763 6 9 2 5 8 1 4 7 0 33733; X by 3, add 3, and cast out the tens. 1 2 3 4 5 6 7 8 9 0 4 7 0 3 6 9 2 8 8 1}5793; × by 3, add 1, and cast out the tens. 1 2 3 4 5 6 7 8 9 06723 0369 25 8 1 4 79713 1 2 3 4 5 6 7 8 9 0 1 X by 3, add 7, and cast out the tens. 9 2 5 8 1 4 7 0 3 67753; × by 3, add 6, and cast out the tens. 36 1 2 3 4 5 6 7 8992909; × by 7, add 1, and cast out the tens. 8 5 2 9 6 3 0 7 4 1 1 2 3 4 5 6 7 8 9 0 1 7 8 9 0 1 9 6 3 074f 4 1 8 5 2 9 6 307 1 2 3 4 5 6 7 8 9 0 1 327 616 54 883r 4 6989; X by 7, add 7, and cast out the tens. 5 296 9999; X by 7, add 6, and cast out the tens. 7 9 887 or ∞ 738 9) 1208 75278; X by 3, add 0 or 5, and cast out the tens. 29248 9 4 8 0 5 1 0 6 0 3 2 8 9 1258; X by 3, add 2 or 7, and cast out the tens. 9298 3218; 7238 X by 3, add 1 or 6, and cast out the tens. 3268 X by 3, add 3 or 8, and cast out the tens. 4 05228; × by 3, add 3 or 8, and cast out the tens. 749 5 20434; by 2, add 4 or 9, and cast out the tens. 7 9 0484; X by 2, add 1 or 6, and cast out the tens. 9 4 4414 0 0 42424 X by 2, add 4 or 9, and cast out the tens. 80∞ 5 9J 4 6 8 0 7 1 2474; X by 2, add 1 or 6, and cast out the tens. 3 2 6 5 9 1 6 8444 4464; X by 2, add 3 or 8, and cast out the tens. |