ENGINEERING MAGAZINE, COMMENCED JANUARY, 1869. Published on the 15th of the month at $5.00 per year. The January number of this MAGAZINE, for the year 1881, begins the Twenty-fourth Volume. Beginning as an Eclectic Journal, and presenting almost exclusively matter selected from current literature, it has gradually become the chief medium through which the leading writers on engineering subjects can best present their original essays to American readers. The attitude of the MAGAZINE has been, and will continue to be, that of a journal of original and selected papers upon subjects relating to modern advanced Engineering. Theoretical and Practical Essays are alike presented in its pages, although the latter largely out-number the former, as best suited to the tastes and demands of the American Engineers. Some of the most valuable contributions to the literature of technical science within the last few years have been first presented in these pages. Among the more extended original contributions to the later volumes may be cited new contributions to Graphical Statics-Transmission of Power by Wire Ropes-Maximum Stresses in Framed Bridges-Momentum and Vis Viva-Rapid Methods of Laying out Gearing-Transmission of Power by Compressed Air-Geographical SurveyingMathematical Theory of Fluid Motion Thermodynamics - Practical Theory of Voussoir Arches-Cable Making for Suspension Bridges, &c., &c. To the above may be added the following valuable essays, translated from foreign sources, which have first appeared in these pages: Linkages and their Applications— The Origin of Metallurgy and The Theory of Ice Machines. The plans for future volumes comprehend many improvements in the same direction. The wants of the educated practical engineer, who desires to keep in the foremost rank of his profession will be steadily kept in view, and our constantly increasing resources for supplying the best of scientific information will be employed to secure such result. Cloth covers for Volumes I. to XXIII. inclusive, elegantly stamped in gilt, will be furnished by the publisher, for fifty cents each. If the back numbers be sent, the volumes will be bound neatly in black cloth and lettered, for seventy-five cents each. The expense of carriage must be borne by the subscriber. Notice to New Subscribers.-Persons commencing their subscriptions with the Twenty-fourth Volume (January, 1881), and who are desirous of possessing the work from its com mencement, will be supplied with Volumes I. to XXIII. inclusive, neatly bound in cloth, for $60.00, in half morocco, $90.00. Notice to Clubs.-An extra copy will be supplied gratis to every Club of Five subscribers at $5.00 each, sent in one remittance. CONTENTS. PAGE THE GEOMETRICAL INTERPRETATION OF IMAGINARY QUAN- THE FLOW OF COMPRESSED AIR THROUGH LONG PIPES. By . Contrib, to Van Nostrand's Magazine... 89 Trans. from Revue Universelle des Mines. 96 THE PRINCIPLES INVOLVED IN THE CONSTRUCTION OF THE THE PROTECTION OF WOOD AND IRON BY PARAFFINE......Builder... J. Kollmann... THE USE OF CEMENTS.. THE PREVENTION OF FLOODS.. COFFERDAMS IN FLOODED RIVERS. By M. Lanteires. ...Engineer.... 115 . 118 137 152 Arm .Papers of Royal Eng. Institute... Abstracts of Inst. Civil Eng..... SIGNALING BY MEANS OF SOUND By E. Price-Edwards...Journal of the Society of Arts. PARAGRAPHS.-St. Gothard Tunnel, 95; History of Zero, 163; Labor Omnia Vincit, 164; Strength of Cylindrical Vessels. 165. REPORTS OF ENGINEERING SOCIETIES.-Boston Society of Civil Engineers; Engineers' Club of Philadelphia: The American Society of Civil Engineers; The Polytechnic Association of the American Institute, 168. ENGINEERING NOTES.-Safety Arrangement for Water Pipes; United States Submarine Surveying; The New Tay Bridge, 170. IRON AND STEEL NOTES.-Customs' Returns of Pig Iron, &c.; New Method of Analyzing Iron; Steel Making in China; Spangenberg's Experiments on the Strength of Iron and Steel, 171. RAILWAY NOTES.-Westinghouse Brake Adopted in France; Breakages of Steel Rails on Russian Railways During 1879; Paper Railway Wheels, 172; Bad Station Accommodations on the London, Chatham and Dover Railway, 173. ORDNANCE AND NAVAL.-Arrangements for the Defence of the Mouth of the Weser, 173; Russian Composite 8-in. Gun, 174. BOOK NOTICES.-Publications Received; Naval Encyclopædia; Steam Boilers, by William H. Shock, Engineer-inChief, U. S. Navy, 174. MISCELLANEOUS.-Boguslawski's Investigations of Deep Sea Soundings; Solidification Under Pressure: The High Towers of Europe; Death of William Lassel, F.R.S.; 176. VAN NOSTRAND'S ENGINEERING MAGAZINE. NO. CXLVI-FEBRUARY, 1881.—VOL. XXIV. THE GEOMETRICAL INTERPRETATION OF IMAGINARY QUANTITIES. Translated from the French of M. Argand by Prof. A. S. HARDY. Contributed to VAN NOSTRAND'S ENGINEERING MAGAZINE. II. Ir is to be observed that while there the conception under which they are exists an infinite variety of directed regarded real. We might apply the lines, practically they are all referred, general term intermedials to all others as will be shortly shown, to KA, KC, which it is not necessary to designate KB, KD, the position unit being KA specially.* the negative KC and the means KB and KD (Fig. 4). It is, furthermore, convenient to classify any two opposite directions under one head, to which we shall apply the term order. The primitive KA with its negative KC we shall designate as the prime order, and the means KB and KD as the medial order. We shall speak of a prime quantity or medial quantity when we refer to one of a prime or medial order, respectively. These terms are derived from the mode of generation of these quantities, and from Vol. XXIV.--No. 2.-7. 8. In accordance with what precedes, we may also modify the language of so-called imaginaries in such a way as to render this part of the subject more simple. In writing +a√-1 or-a√—1, we indicate explicitly the way in which the quantity is generated, which in certain cases may be useful; but ordinarily we leave the mode of generation out of consideration, and -1 is only a particular kind of unit to which the number a is referred. It is, therefore, not abso lutely essential to keep the mode of generation in view. Again, the expression av-1 shows V-1 to be a multiplier of a; but really V-1, in av-1, is no more a factor than is +1 in +a, or -1 purpose to propose the substitution of the nomencla *It has been already remarked that the relations said to exist between lines, when we take their directions into account, cannot as yet be regarded other than hypothetical. It is, therefore, very far from our ture above described for that commonly employed; but to make use of it only because, in general, it is desirable to avoid the employment of terms whose real meaning is at variance with the ideas we wish to are concerned with an express, even when we hypothesis. ~ that they are contrary to usage, and their trial, at least, is permissible. Fig. 5. K Q S in -Ɑ. Now we do not write + 1. a,1. a, but simply + a, a, and the sign which precedes a itself indicates 9. We are now to examine the various what kind of a unit this number ways in which directed lines are comexpresses. We may then apply a bined by addition and multiplication, similar method to imaginary quanti- and to determine the resulting constructies, writing for example a and xa tions. Suppose, first, that we have to instead of +av-1 and -av-1, the add to the positive prime line KP signs andy being_reciprocally posi- (Fig. 5) the line KQ, also a positive tive and negative. To multiply these signs, we observe that either multiplied by itself gives, and, consequently, multiplied by each other they give +. Moreover, a single rule, applicable to any number of factors, may be estab- prime; the construction would not differ lished; let every straight line, horizontal from that of finding the sum of the or vertical, in the signs to be multiplied, absolute lines KP, KQ; it consists in have a value 2, and every curved one a laying off the distance PR=KQ on the value 1; we shall have for the four signs prolongation of KP. We then have the following values: KP+KQ=KP+PR=KR. To add a ~=1, −=2, x=3, +=4. negative prime line QK to another PK, Then take the sum of the values of all the construction is the same, but in the the factors and subtract as many times 4 opposite direction, and we should have as is necessary to make the remainder PK+QK=PK+RP=RK. In general, one of the numbers 1, 2, 3, 4; this re- if we are to add two lines of the mainder will be the value of the sign of the product; and so, for division, sub- same direction, AB, AC, we take in AC, tract the sum of the sign values of the this direction, PQ = AB, QR = divisor from that of the dividend, hav- and we have PQ + QR = AB+ AC=PR. ing added if necessary a multiple of 4 to If we are to add to the positive line KP the latter, and the remainder will indi- the negative QK, we take a distance cate the sign of the quotient. It is to PS-QK in the negative direction from be noticed that these operations are P, and obtain KP+QK=KS=QP. The those of multiplication and division by logarithms; this analogy will be brought order. same course is pursued for any other more fully into view. Now, the principle underlying these These new signs would abridge the constructions is that we regard P, the notation, and perhaps render the calcu lus of imaginaries more convenient, final point of KP, as the initial point of errors of sign being sometimes easily the line to be added, and that we take made. We shall employ them in what respectively for the initial and final follows, without implying on that ac- points of the sum, the initial point of count that they should be adopted. KP and the final point of the added Doubtless to every innovation, even a line. Applying this same principle to rational one, there is an intrinsic objec- lines of other orders, we conclude that tion; but no progress would be made if K, P, R, being any points whatever, we they were rejected, for the only reason always have KP+PR=KR; and as each of the lines KP, PR may also be the sum of two lines, as KM+MP, PN+NR, M and N being arbitrarily chosen, we conclude that, in general, A, B; M, N, O, . . . . R, S, T being any points The quantity m±n V-1 being denoted by m~n, or by myn, the single sign, or x, replacing the four signs +, V -, 1. † For example, let it be required to multiply -m Vc by+n-ca. The product of the two coefficients is -mn; that of the two radicals is -cd; and the final product is +mncd. In the new notation the two factors are ~mve, xn Ved, or xm Vc,~n Vcd, and by whatever, AB=AM÷MN+NO+O... the rule we at once obtain+mned. This advantage- + if it be one-would not exist for an experienced calcu + lator, who by a simple inspection of the factors would points A, B, M, R+RS+ST+TB. The may coincide or read the product; but not every one possesses this be so situated that the lines AM, MN,.... faculty. |